Step one for me and I set-up three guitars already today is to sight down the neck and set the truss rod and relief.
These Teles have Fender style bolt on necks which are notorious especially in a bit of time for ski ramping at the very end over the body. This is the single most common reason for fret buzz on Strats and Teles regardless of the maker. Seems bolt on necks do move in time sometimes and I'm not sure if RH has anything to do with it.
So check for a ski ramp where the remedy is to impart something that I speak of here more often than some want to hear, fall-away. Or in other words your faith in this fret work may be misplaced and stainless frets actually are more likely to have issues because the person doing the leveling gets tired.... and stainless is difficult to work with.
So I would be first looking for a ski ramp after changing the strings as Chris suggested.
Next would be the Tele saddles. If the bridge is slightly off center a string could be contacting an adjustment screw on one of the saddles. This is common too with Teles which is part of the reason people like me will pay big bucks for Calham saddles or a few others such as Wilkinson.
If you can fret in any position and it still buzzes it not the nut.
So set-ups are sequential where one thing we do builds on the next thing we do and when I have one like this and I can't see for sure what the issue is I simply do my set-up sequence and like fishing with a net it always catches it and resolves the issue. We approach set-ups sequentially because each thing we set this way never needs to be revisited after doing other things down the road AND all dependencies are addressed.
1) Set truss rod and relief
2). Cut nut slots (which even though I don't suspect them I guarantee you that they are way too high) and they need to be cut before setting action since they change the action.
3). Set action at the 12th and play every note with a moderate attack unless you know the steward to be heavy handed then hit harder to test it out.
4). Set intonation.
5). Play Purple Rain and annoy Dave.
So that's what I would do and again I would not have any confidence in the fret work of pretty much anyone. People don't bring us stuff because it's working great....
One last thing. The action expectation what is it? The guitar should be able to play with a moderate attack buzz free with the high e at 4/64th" and the low e at 4.5/64th" both set at the 12th under string tension tuned to pitch and in the playing position. If you can get it buzz free at these specs anything lower likely requires first class fret work to make that happen and is outside the scope of a professional set-up and another job billing wise.
Our shop is a bit on the cool side now and I'm old as dirt so I'm always cold and like your sweater