First of all, Welcome to the Horse Forum!
I have to admit that I have never heard of a 5 stage vetting. This may be total ignorance on my part, or it may be that it is an accepted procedure in your area, and not in mine. I don't know where you are from, or how you intend to use your new horse.
If I were buying an expensive horse that I intended to use in a demanding discipline, I'd certainly perform a more extensive vetting than if I were just buying a horse to hack around the neighborhood! I can't answer your question about insurance because I would assume it would depend upon what type of insurance you are getting. A general policy to cover for colic surgery would be far different from insurance designed to make sure the horse was usable, or even life insurance. This is a question you need to ask your insurance agent.
I have returned 2 horses that failed their vet checks. Each time, it was a problem that would directly affect their ability to do what I wanted them to be able to do! That is the key....you have to consider WHY they fail their vetting. Make sure you know the vet doing the exam and that you have clear lines of communication with him. He has to have a very clear idea of what you expect from the horse.
Flexion tests are not difficult for a sound horse to pass. A horse either has decent joints that move correctly or he doesn't. There are no areas that are more common for a horse to fail, because each horse is an individual, and each potential owner has different expectations. An older horse that is intended for a child's first horse may have problems that would totally cause him to fail an exam for a field hunter-jumper...but he may be perfect for a child to learn on. You need to talk with your vet and determine what problems would be deal breakers.
Your vet will be doing most of the "looking." If you've reached the point where you're paying a vet, then you must have already determined that the horse's size, temperament and training level are what you are hunting for. If your vet finds ANY problem, he should discuss it with you and explain why it's a problem and what options you have for treatment. You can look at how the horse handles the exam...how he tolerates being examined by a stranger. It shouldn't be an issue for him.
Your vet will be physically examining the horse. You should have already ridden the horse. I doubt you'll be asked to ride him. Depending on whether or not your vet brings an assistant, you may be asked to lead the horse at a walk or trot...or the owner may do this while the vet observes. I was totally shocked a couple years ago when my vet rode the horse I was considering....bareback with a lead rope! That is above and beyond what is expected!!!
You don't mention what breed you are looking at or where you are, but one thing I would SERIOUSLY consider if I were spending a lot of money on a horse (and maybe even if I wasn't) is genetic testing. By analyzing hair strands, you can determine if your horse has any inheritable diseases. There are at least 5 that will show up in these tests: HERDA, GBED, HYPP, MH and PSSMI. Some of these can be moderated by a special diet, and others can't. Either way, you do NOT want to buy a horse with these problems. If the horse you are considering is young, its current owner may not be aware that he has any these problems, so it's not like he could tell you. These problems can't be seen by just looking at the horse.
Good luck with your new horse!