So, I had a species that primarily lives off meat and insects, but I came across numerous articles which seem to point out that sapience and civilization is near impossible without being omnivorous. This led to me wondering, Based on our current understanding of how sapience evolves, are insectivores and carnivores viable to become sapient, create civilization, and get technology to at least the level of the Middle Ages over time? What complications would arise from this? Pros and cons?
Those articles claiming that sapience and civilization is near impossible without being omnivorous are (perhaps) suffering from anthropocentrism: "The only evidence we have is ourselves, so the only possible answer is like ourselves."
You wrote that your species "primarily lives off meat and insects" (emphasis mine). So that means they do eat things other than meat and insects... they are omnivorous! So, if you want to be in line with those articles... you're golden.
Either way: remember that sapience is an evolutionary adaptation, not necessarily an evolutionary advantage. In other words, being sapient isn't necessarily better or worse / more evolved than not being sapient.
First, a clarification: "Sapience" is the ability to know things, and reason with that knowledge.
Are octopuses sapient? It certainly seems so. What about chimpanzees? Yeah, I reckon. Dogs? Uh huh. Elephants? Sure. Dolphins? It can be argued, indeed. What about horses? Hmmm. Again, it could be argued.
No one really knows what environmental pressures result in sapience, but going by the definition and the examples I just listed off the top of my head, dietary choices and even environment don't seem to matter. So your bug and meat eating critters are fair game.
Now, what about civilization?
Well, what is civilization?
Wikipedia says, "A civilization is any complex society characterized by urban development, social stratification, a form of government and symbolic systems of communication such as writing."
Good enough. So while there might be sapient complex societies with social stratification (see, again, chimpanzees), that's not a guarantee of "civilization" as we define it.
However! Do elephants need writing? Nah. But they do have ultrasonic long-distance communication and seem to have institutional (cultural) memory (as do ravens, btw). So maybe "oral history" is just fine rather than writing.
Urban development and a form of government are, essentially, technology with the purpose of scaling up small groups. Where there is prosperity (read: agriculture or a perennial and abundant food source) there is population, and with population, urbanization and government are handy (perhaps necessary) tools with which to manage a large group consistently and with a minimum of conflict... things advantageous to the ultimate goal of any species: to make sure one's offspring have offspring.
That's a very long winded way of saying, again, sure your bug-and-meat eating critters can have a civilization, given an environment with the right resources in the right amount.
You can extrapolate the complications over time by considering the behavior of your proto-critters, much as you can extrapolate human culture and behavior by looking at the behavior of "lower" primates. Think about how elephant society works and draw conclusions as to what they might be if there were millions or billions of elephants instead. That's an extreme example, but I hope you get my point: an elephant civilization would be very different from a crow civilization would be very different from an octopus civilization.
As for pros and cons... evolution doesn't think in terms of pros and cons, better or worse. If a species' babies grow up to have babies, that species is well-adapted to their environment. If sapience means more babies grow up to have babies, that might become a dominant trait. If the population gets big enough and language helps them stay organized and have grandkids, language (or agriculture or animal husbandry or roads...) might become a dominant trait.
Go forth with your bug and meat eating critters, and multiply. :-D
The real issue here is a consequence of Trophic Levels and the Ten-Percent Law. As quick summary, significant energy losses occur every time an organism eats something, so it's getting about 10% of the energy that the thing it ate got from it's source of energy.
Since every creature has some energy needs and civilization needs a lot of creatures in a (relatively) small area, you need a lot of resources to support these insectivores/carnivores. This is why you have a lot of primary consumers (like insects who eat plants) and much fewer tertiary consumers (like tigers).
This means that carnivorous civilizations are limited to areas that consistently produce a lot of food or are constrained to move with their food source(s). This isn't to say culture or technology is impossible while being entirely carnivorous. The Inuit have a nearly all-meat diet and are still a civilization. There are social groups with dolphins and whales in spite of their all-animal diets.
There is also the idea that more advanced technologies happen when people spend less of their time and effort simply trying to get food and more time and effort on learning about and experimenting with their world. This is simply harder when you are not a primary consumer or producer.
Many plants are poisonous or only grow in certain places at certain times of year; so, herbivores need a good memory and discrimination skills. Carnivores on the other hand benefit from complex reasoning skills and a certain sense of self-awareness to be effective at tracking, stalking, and ambushing their prey. The reason Omnivores are normally associated with intelligence is that they have both evolutionary pressures so they can develop memory, discrimination, complex reasoning, and self-awareness all in one package. These are the general building blocks of sentient thought.
That said, it is possible for any animal to find a niche in which all of these skills are needed regardless of what they eat. For example, let's say your insectivore were some kind of migratory animal that survives by moving between different seasonal insect blooms that happen at different times of the year and in different places, they would need a good memory. If they lived in an environment where some of their prey insects use mimicry to closely resemble poisonous bugs, then they could develop a strong sense of discrimination to keep them from eating the bad bugs. If some of the bugs they hunt are faster than they are, then they would need to develop complex reasoning to figure out how to ambush their food, and if some of the bugs they hunt have keen senses or if they are a prey animal of something with keen senses, then they would need to develop to be self aware enough to not make a lot of noise, or stalk without cover, or stand upwind of things they are stalking/hiding from.
This just leaves the final element of being driven to modify one's own environment for better survival. For humans, it was agriculture that really sparked the growth of civilization, but not all animals that modify their environment do so for growing plants to eat. Some animals build nests or burrows to keep their young safe. Some animals build caches for storing food for winter. Various species of ants farm fungi or honey dew secreting insects. And let's not forget, humans raise animals for food too. The ancient Steppe civilizations produced great empires like the Mongols and the Scythians off of animal husbandry and not agriculture; so, your animals may get into the practice of farming prey animals as they get smarter so that they can stop wasting energy on migrating. Once they are "farming" for food, their evolution could follow a similar growth as humans.
I don't forsee much of a problem in evolving intelligence/sapience, but not eating plants makes it much harder to live by farming. You need a lot more land, or maybe very fertile land, to produce the same amount of calories from meat instead of plants. That makes it much more difficult to move from hunter/gatherers to farming communities, which are needed for a Middle Ages level of technology. You can't really take your iron bloomery with you every time the community moves with the herds.
However farming insects could solve this in part. In modern insect farming, some species can produce 50% of the food mass as insect meat (Wikipedia link), so then you only need twice the land area compared to eating plants directly. Farming insects presents some difficulties of their own. Unlike plants they run away, but they can't be stopped with something like a fence, and I don't think there are any insects that can be herded. (Though in your fictional world there might be. Nest building insects like bees and ants tend to stay in one place.) So I guess the bugs need to be kept in some kind of boxes, with the 'humans' harvesting the vegetables and feeding it to the bugs.
On Earth, for mammals, one of the potential problems is that intelligence is expensive, and requires omega-3s, which can be tricky to find in nature.
Insects are apparently a potentially good source of Omega-3s (per https://www.infona.pl/resource/bwmeta1.element.elsevier-c89c5b3e-6a99-335e-94ad-4c556add6141). So, if you're in a place where there's a LOT of insect biomass, you probably have the nutritional basis to support evolutionary pressures towards a big brain.
Note that large brains aren't always required for intelligence: it's just the cheapest way to get smart. Turns out, because weight is a huge issue when flying, Corvids and similarly smart birds do much more with far less brain, by instead having their brains loop the information more often. This is slower in theory, but since their brains are smaller, they save on signalling transit time, and achieve intelligence on a par with animals that have a FAR higher brain:body mass ratio.
There are also distributed neural systems, such as the nine brains of an octopus, where the brain is scattered around in the body.
And there's hive intelligences, where the "intelligence" is an emergent property of the complex behaviors of the members in response to pheromonal stimuli. I'm not convinced that there's a way for hive intelligence can result in anything approaching the lateral-thinking and symbology of true sapience, though.
So what kind of evolutionary pressures support large brains, or one of the alternatives? Well... for the most part it seems to be about having to think up new approaches to finding resources, like shelter, safety, or food.
If you can make the insects plentiful, but variable (different insect types in different seasons? Each species dangerous in a different way, found in a different environment, etc?) so that the creatures have to think adaptively and cooperatively to forage for them, that could work.
The key trigger for sapience is fire/cooking. Raw food requires a longer gut to get the maximum energy from eating. Cooking starts the breakdown process before the food is consumed.
What this means is the creature requires less intestines and the intestines require a lot of energy to run. Cooking the food allows a smaller gut and creates a surplus of energy in the body which allows evolution to start increasing the brain power (which also takes a lot of energy)
As long as there's enough insects for them to cook, it's possible.