I suppose some deductive reasoning may assist with the answer.
Simple question: do any of your other AF-S lenses do similar things.
As AF-S lenses use their own motor, the two most likely culprits for causing such an issue are:
1/ the lens itself .. a possible warning that the SWM motor in the lens is on it's way out. (they can die)
2/ As the lens takes instructions from the camera on when to focus, maybe (as you already indicated) cleaning the contacts will help.
But if you think about it in a logical manner, if the contacts from camera to lens(for both AF signal and power) were dodgy in any way, you'd more likely get randomly occurring focus issues, rather than that you described where durability is the issue.
Think of it in a similar vein to your car engine overheating. If you place a load on it, it starts to heat up and it will either stop, or you have too.
As for cleaning contacts, both lens and cameras .. I use a clean pencil eraser. Lightly rub thee contacts of both devices.
I've had some issues with some lenses and it's worked well when they played up... always the lens side of things.
So, as your first point of attention .. you really should be testing a similar focus regime with any other AF-S type lens .. if you have one.
If you only have AF-D(or AF) type lenses ... ie. screw driven types .. don't bother. It may reveal no significant result, or may produce a misleading result.
The screw driven lenses use similar but still different enough focusing systems to make this an important point of difference.
Even if the lens is a HSM(Sigma) or USD(Tamron) type lens(or any other that relies purely on an electronic connection between camera and lens for focus.
So if you do test any other lenses and they're operating fine, then obviously the 70-200 is the issue(I reckon most likely the cause).
If any other electronically driven lens is having the same or similar issues, then obviously it's the camera that's the problem.
For your sake, pray it's the former and not the latter .. except if it's just a contact (on the body) cleaning issue on the camera.
FWIW: My only two issues with respect to camera lens communication have been:
1/ my D300 and Sigma 10-20mm lens(HSM). Camera simply wouldn't recognise it one day (totally out of the blue).
Mounted unmounted remounted the lens and still naught. This was while I was out and about, so of course the lens had to be removed and another lens was mounted for the rest of that trip.
On the way home on that trip whilst I was just driving, it suddenly occurred to me .. AH! the contacts!
Soon as I got home I cleaned the contacts on the lens, after first confirming that it was still a problem and ...
(that lens is still going strong 7 years later)
and
2/ The other issue was the Nikon 18-105VR kit lens. A total focus failure. Once again didn't initially think the contacts would be the issue, my immediate reaction was a failed focus motor on the lens.
This one was still fairly new .. less than a year old. I think it's natural to assume the worst in such a situation.
This was at home, and after a very quick partial dissassembly of the rear mount .. again the thought came to me that it may have just been a contact issue. Put it together, cleaned the lens contacts .. and
again.
Both times I just used a decent quality soft-ish pencil eraser on the lens contacts .. just lightly .. and all good.
Hope it all works out for 'ya.