It is varietal specific but is also influenced by environment.
I have a Tannat vine grown in the heat of North Alabama. The grapes will coat the tongue like a green persimmon but it doesn't make bitter wine like it does in cooler places. There is much study about seeds tannins vs. skin tannins and then wine tannins. Good info to google.
My experience is very limited. I've been told by a vineyard owner that Tannat stands for "to not drink". I've tasted a Tannat from Uruguay that was a bit harsh. I made a gallon this year and was surprised at how drinkable it MAY be. After the first rack, it isn't bitter but is thick and has a nice fruit quality. The jury is still out.
Yeah, my understanding is that tannat can be hit or miss. Besides being very tannic, I've heard that its also very acidic. Most of the wineries here in IN use it for blending with less tannic varieties. The best blends I've tried were blends with cab franc, another variety I think I can get away with. I say that because I'm in a location that makes vinifera very tempting to try growing but the time investment and the climate i'm in makes me equally hesitant.
I have read it is grown commercially in Virginia which can have fairly cold winters. Of course we haven't had a brutal winter for a few years now. You might pose the question to Double A vineyards in NY. I ordered a single Tannat from them four years ago.