IMO, the fullbacks' responsibilities (in order of importance) in Klopp's system are:
1. Offensive. Direct play, providing width and additional numbers in attack. Involved in set plays on their strong foot side.
2. Transitional. High-press/intense defensive pressure in the attacking + middle third when losing possession.
3. Defensive. Recovery/dropping back into the back-4 defensive posture. Typically, 1 or 2 of the CMs (Wijnaldum, Fabinho, Henderson, Milner) will sit deep to cover for the fullbacks, and the 2 CBs will split wide. Klopp likes mobile, defensively solid CMs... the more attacking minded box-to-box types struggle in his system (Keita, Oxlade-Chamberlain).
We may be arguing about semantics here. I just don't think that what TAA + Robertson and what City's fullbacks are doing is "crossing" in the classic English sense. I mean, Pep has smashed transfer-records and gone through about 10 world-class fullbacks (of all varieties) trying to find one that works in his system. Often settling for Zinchenko (LMFAO).
As for quality "crossers" that I'd like to see at Spurs, I'm a huge fan of James Ward-Prowse (criminally under-rated) and Dwight McNeil (we should be all over him if Burnley get relegated).