Be aware that a 16 year old pony is likely to at least fail the flexion test part of the vetting. So, essentially fail a vetting.
Depends on cost. We've just bought a 21 y/old who was fat, unfit and quite grumpy to be honest. I suspected Cushing's too (he's tested negative). £1k. I did know his owners and I had seen him ridden enough to see how responsive and easy he was though. He just needed work.
Regular work, weight loss, having a bond with one particular person (DD) and making sure he's kept up to date with dentist, farrier, physio, worming and all, you wouldn't recognise him. A true school master of a pony to ride, said with honesty. Fast though, lord knows how fast he was at 10 
I am by no means the expert, but a lot of friends are, and say his movements are clean and smooth, no hint of stiffness.
We've been lucky though. We haven't in the past. Maybe have the vetting, for the sake of a few hundred quid?