Be so careful with the vet check.
Yes, do not use the barn vet.
But, here's vet checks IMO:
Over the age of about 12, most horses will not pass a basic vet check. Shoot, I wouldn't either with the way the joints are pulled on.
After the basic vet check, it is recommended that pictures be done on any suspect joints.
Okay, this is where it can get ugly. X-rays cost a lot of money, well beyond the basic vet check. 1st: you still have to pay for the x-rays even if you don't buy the horse. 2nd: the pictures can show ambiguities AND the vet won't tell you if the horse is going to be sound or not.
Where it can get uglier. Vets won't guarantee a horse for you. I don't blame them, we're talking horses here.
Why/How: A horse can show signs of arthritis very early and it may not affect anything until way later. For example: 10 year old horse, may not experience any problems until 25.
Also, a horse can come up positive on joints, negative on pictures and go lame on soft tissue.
So, if you have been leasing this horse and riding the horse quite a bit, you can probably rule out drugs (unless there's a chance the person you are leasing from is buting the horse before you ride).
If you have been riding the horse for a while and the horse has been fine, then you can most likely consider the horse fine.
You could buy this horse and the next day s/he could break a leg. It's just a possibility.
Go with your gut.
If the horse apprears healthy, does not need anything special in that way, and you are comfortable with the condition and conformation of the legs, then 1,2 3 go.
If you need training on this horse to suit you, or training is suggested, then it is NOT the horse for you.
Lessons, absolutely. Everyone could use a lesson. IMO. If the Lessor is planning to show/ride your horse after you buy, ask yourself WHY.
Anyway, good luck with all of it. You seem like someone who has gone over this horse-buying a little bit and it seems like a good idea.
Horses are amazing, wonderful creatures. They are a beneficial part to humans, IMO. But, the industry is unregulated so keep your eyes wide open. Vets, farriers, chiropractors, massage therapists are regulated by some, others by few. Trainers, instructors are regulated almost never.
Good luck and good for you for asking.