I would call your horse just hitting mid-life, not even close to OLD. Unless he has been used very hard, kept in extreme condtions, like year around range horse herd, previously injured, he should not be having old-age problems yet.
I would agree that the Vet look-over with Spring shots should be enough at this point and for a while yet.
Mine didn't start slowing down until about 23-25 years. And our family are the only ones who would say horses had "slowed down a bit", no one else would notice it. This would be maybe not doing the 20 mile trail ride sections daily, just doing 10 or so miles a day. Maybe only enter 10 Pleasure type and speed classes at 4-H show instead of 15.
We find that an "aging" horse who is kept fairly fit, used regularly during the week, stays younger, sounder, recovers from use quicker, than the same age horse only used lightly. And some regular gallop work every few days, makes a HUGE improvement in their body systems, a real plus in their fitness program.
So I would consider a 16yr old to be in the prime of life, ready and able to do any request in the riding catagory I felt like doing. Not ready to be retired to light use yet!
Is there some age when it's a good idea to have the vet out for a general check to see if all is well?
Sure. From birth on up. I think it's a good practice to have your vet be familiar with your horses, your place, your style of management, etc. I have mine out spring and fall to do general checkups, shots, teeth, whatever . . . the "health maintenance" part adds almost nothing to the bill or the time spent, but it fosters the good relationship I have with my vet and allows me to count myself among his list of loyal clients. So when I need him at 2 in the morning . . .
My Gwennie was sixteen when I *bought* her. She very quickly took my Novice butt up to Preliminary/CCI* level and kept it there for several years! Sixteen can be a great age for a horse, if they've been looked after properly, are kept fit, and good vet care is a part of that.
Mine get check-ups every spring along with shots. It usually includes some bloodwork (CBC and chem panel), a jog/maybe flexions, listening to heart/lungs, checking vision, palpating everything- maybe some basic manual neuro tests if horse has a history.
I call it the "old man exam", and both of my older horses (17y & 24y) get annual bloodwork and a check up in the spring. It's more for baseline, so that when something does happen, the vets know the horse(s) and have "normal" bloodwork on file (we found some parameters for each horse normally run high/low, which is good to know for the future). I don't do flexions/lameness exams, but they do palpations, optical exams, nutritional analysis/recommendations, and then I get to ask the non-important questions that I've been building all year.
He does get a quick check with shots/floating and my vet has been his vet since he was on the track. He's known my horse longer than I have. Plus, he's the vet for most of the horses at my barn so he's often out there for something or another (27 horses).
This spring we'll do a bit more than the usual quick check.