Its easier to get an armored vertebrate than a giant arthropod.
You see: arthropods are remarkably successful species, their exoskeleton provide both safety and structural support, and allows them to be nimble. Problem: exoskeletons work at their best on smaller animals. Insects are so small they Ca afford to transmit oxygen directly to their tissues through a tracheal system. This is a remarkable effective way to ensure your tissues have all the oxygen they nees...as long as you're small enough. At bigger sizes this becomes less and less efficient to the point of unsustainability.
As for their exoskeleton, its the same song and dance: Imagine the following: you just hit puberty, and you're going to grow into an adult. Insects do it through molting, shedding their exoskeletons and growing a new soft layer underneath them, which will slowly harden until it works as well as the previous one, except during this process the insect can stretch this layer while it’s still soft so the final product it bigger. This process has some advantages like allowing the bug to be fully armored for the rest of the time and even do things like regenerate lost limbs in the case of some arthropods like crabs, but the downside is that you either stop growing or you'll have to undergo a stage during which you have no support for your body or muscles and are completely defenseless. In addition, if anything goes wrong during it you will most likely die. Look at coconut crabs for example:
(in here we have an adult coconut crab compared to a trash can it is likely trying to break into in search of food)
These guys are Massive for an arthropod, being the biggest terrestrial arthropod on earth, but the process needed to reach this size is extremely dangerous, and each subsequent molting (which in these guys can take up to three months to conclude, happening about once a year) becomes more troublesome as they grow, until they reach a point where their molting fails, assuming nothing gets to them before that.
Coconut crabs also show us another problem about giant arthropods: they "only" get as big as some dogs, but if there's something these guys are certainly not famous for, its their speed, these guys couldn't go fast to save their shells (litteraly). Much like you said, a giant arthropod would require a massively thick exoskeleton and legs to support it, meaning that even if it did have lungs (which fun fact: coconut crabs have something similar called brachiostegal lungs) and a closed circulatory system more capable of transferring oxygen and nutrients, it would most certainly not break any records of speed. If anything, it would probably work closer to a giant turtle, relying more on its armored hide and size to evade predation rather than speed.
So basically: giant horse-like beetles with exoskeleton really cannot work well on earth-like conditions. Its important to remember arthropods aren't just older than mammals, they're older than sharks, trees and the rings of saturn. If they could've successfully occupied the niches of large animals they most likely would've, but even the largest land arthropod ever, arthropleura the giant millipede, was relatively low to the ground, even though it was 2 meters long. In other words, your swamp habitat could be a massive danger,as its slowness and great weight could cause it to get stuck and drown or fall prey to other problems like starvation or becoming easy prey.
So, what's the second best option? To cheat, of course!
Armored vertebrate bettle
Exoskeletons work great for small animals, but for things closer to horse sized, exoskeletons are out and endoskeletons are in. a vertebrate creature has a much easier time sustaining its own weight at larger sizes than an arthropod, and the fact its bones are all enveloped by living tissue means they can grow without annoying, dangerous molts. Better yet, we have several historical cases of endoskeletal having animals that are horse sized and even bigger than that, so its certainly plausible.
But quadrupedal vertebrates are no fun, we want bug-like animals with 6 legs! what then? well, one of the likely ancestors of all land dwelling vertebrates was a fish with six fins, four of which would later I've way to legs. With some tempering around the condition to make it so six limbs became more advantageous we could easily make it so they became the norm. Tha way we have six legged creature that can grow to be horse sized. Add in some environmental pressures for it to develop bony armor first and later to grow larger and you could end up with something somewhat reminiscent of a cross between a horse and a giant beetle. However, there are problems:
See, here is the part where a choice must be made: bettles are small, so in their case we can still have things like the tiger beetle, which is still armored, capable of flight and capable of running so fast it physically cannot process what's its seeing while it is running. For something as big as a horse on earth like conditions however, we reach an issue:
- if its big and armored, it can't be very fast.
- if it is big and fast, it can't be very armored.
- if it is fast and armored, it can't be very big.
These traits in general tug in different directions:
- larger animals can take bigger steps and so can move faster, but they also have greater inertia and are heavier than smaller animals, and are more limited than smaller animals in being both well armored and fast.
- Moving fast requires many adaptations, but one of them is your weight and how it is distributed. an animal specialized to move fast needs longer legs for greater strides, with most of their weight concentrated near the body.
- Using armor meanwhile means attaching mostly bone and keratin over yourself to keep you safe. In addition to being metabolically expensive, good armor will rarely be light, and armored animals benefit more from shorter legs and a lower center of gravity so they can focus their armor on the upper parts of their body and be hard to topple over, with limited use for great speeds.
So while you could have our six legged horse to be well armored, it will also fall in line with the coconut crab scenario: the more armor it has, the heavier it will be and more pressures for it to be slower and down to earth will exist.
Meanwhile, a six legged horse bettle that is fast and with longer legs would have to have its armor be mostly a lighter layer leftover from its evolution covering its body, like a lesser version of an armadllo's osteoderm armor, meaning they'd rely more on outrunning predators than tanking blows, with some species maybe keeping specific parts of their body more well armored while leaving the rest mostly vestigial, likely at the cost of some speed for some defense or to impress mates.
There's another issue here however, and its the swampy environment: horses and other cursorial animals (animals adapted to run far and fast) usually evolve in plane regions with large open fields where the only real options of travelling are to either walk/run or to fly. Swamps meanwhile favor heavily either smaller lighter animals that can properly distribute their weight to not sink or semiaquatic animals that can walk on land and swim in the waters as far as I understand. In other words: much like you don't often find horses living in swamps, you're not likely to find your horse beetle living in one either.
I would discuss flight here too since azhdarchid pterosaurs show us that something the size od a giraffe can still be able to fly at least for shorter distances with the right adaptations, but in general making armored horse sized hexapods would most liekly require them so basically sacrifice their armor and a pair of limbs for the sake of being as light as possible, and even then I doubt it could be used as a flying mount on earth-like conditions safe for very specific cases, and even then likely requiring selective breeding to have animals with very long and broad wings (assuming membranous wings because feathers don't work as well at these sizes) for a greater ability to support weight and require both mounter and gear to remain at a very limitied and most likely very small weight limit.
