I'm afraid your idea of genetic engineering is incorrect, but in a way that is pretty standard.
Genetic engineering is any modification to the genome/DNA of an organism to give that organism a desired effect.
Genetic engineering is the process of using recombinant DNA (rDNA) technology to alter the genetic makeup of an organism.
That definition doesn't mean it's always done in a lab. Humans have been practicing "genetic engineering" for millennia by cross-breeding different species of the same type of organism. Cross-breeding isn't a direct modification of DNA, but it's still modifying DNA in a way that is considered best for humans. For example, the wheat we eat is nothing like the wheat the original farmers grew.
However, this wheat was very different to the crops that fill our fields today – the ears would tower over our modern dwarf varieties, commonly reaching 160 centimetres tall, and with great genetic diversity. These ‘landrace’ varieties were created by generations of natural selection and farmers saving diverse seed year after year.
This happens with dogs, cats, and all kinds of other animals.
A dog breed is a particular strain or dog type that was purposefully bred by humans to perform specific tasks, such as herding, hunting, and guarding.
This happened with cattle as well. So really, the cattle we have today is already genetically modified.
Holstein cows originated in the Netherlands approximately 2,000 years ago. Two breeds of cattle, black animals from the Batavians (present-day Germany) and white animals from the Friesians (present-day Holland), were crossed to create a new breed of cattle. This crossbreeding led to a high milk-producing animal that was able to do so on limited feed resources.
"... they replaced all of their domesticated cattle for one single genetically engineered species that is designed for being cattle."
Really, that sentence doesn't make sense. Today's cattle are already "genetically engineered" for being cattle.
The misconception you and so many other people have is that genetic engineering means you start with nothing and create something completely new, but that's rarely the case. Most of the time it's just a minor modification of an existing organism, even in the lab.
The Plasmid Method -
This method is the most commonly used method in genetic engineering. This method uses small circular pieces of a DNA molecule called plasmids. This method is mainly used for altering microorganisms such as bacteria.
The Vector Method -
The vector method uses techniques similar to the plasmid method. This method uses vectors, which are small carrier molecules, which are normally viruses. Viruses are made of a protein capsule and have their DNA inside, they attach onto a cell then inserts its DNA or RNA into the host cell, then it detaches itself. The DNA, now inside the host cell, will start replicating itself by using the genetic information of the host cell, which means the gene that was inserted will now be part of the host cell.
The Biolistic Method -
The biolistic method is also called the gene gun method, and like its nickname this method uses a gun. This method is mainly used for the engineering of the plants, although the science is evolving to do animals as well.
In a way, genetic engineering scientists are lazy. They want to start out with something that's as close to their desired goals as possible and change as little as possible. But in every other way, these scientists are very much not lazy. They are actually more concerned about dangerous modifications than the "average Joe". They don't want to kill anyone, cause allergic reactions, have the new breed "take over the world", and all the other horror stories you hear in sci-fi and politics. Not only is this a horrible outcome, but anything like it would likely also cause them to lose their jobs, as well as extremists against genetic engineering trying to end it completely.
And really, they want to change as little as possible so they have more control of the process and have less probability of a bad outcome.
But what you might be thinking about is using the best parts of multiple organisms to create a "super cow". This would be cattle that can do everything current cattle do, but better. Cut off a leg and it grows back. Hook up a milking machine for a constant stream of milk. Daily or weekly calving. Fewer "cow farts". And do it all on less food and water.
There's quite a few problems with all that, and the most significant are probably not even technical. It's a problem of human perception. There's already quite a lot of push back against GMO foods (and this and many more). There's also still a lot of education needed for a large amount of people to even understand what genetic modification really is, how much it's tested, and all the other things that go with it. Questions like yours is one indicator of that need. I'm sure I've made mistakes in this answer, which also goes to show that more education is needed.
When asked which of three positions best fits their viewpoints, about half of Americans (48%) say the health effects of GM [genetically modified] foods are no different than other foods, 39% say GM foods are worse for one’s health and one-in-ten (10%) say such foods are better for one’s health.