Which site for puzzles would be best chess.com, lichess, chesstempo? I'm currently using chess.com, but I notice that the puzzle ratings are inflated and sometimes you wouldn't even think that they would happen in a real game, while some don't even work. This question is just an opinion.
I think chess.com is the most inflated out of the three, then lichess, and finally chesstempo. Chesstempo has the hardest puzzles by far. Maybe that's your best bet, but I thinkchess.com puzzles are fine too. I don't like lichess puzzles because they are sort of computerized.
As for your other remarks, even though they won't all appear in your games, they can still train your calculation, which is why I think studies are great too. I'm pretty sure that all the puzzles work as well. You're probably just missing something.
Which site for puzzles would be best chess.com, lichess, chesstempo? I'm currently using chess.com, but I notice that the puzzle ratings are inflated and sometimes you wouldn't even think that they would happen in a real game, while some don't even work. This question is just an opinion.