I think the value of tactics training for me is that it forces me to calculate accurately. Some puzzles are far better at that than others. Like OldPatzerMike, I am only to well aware that my calculation skills are suspect.
Also, over time you do learn patterns and you may recognize those in your games. Gradually I am recognizing opportunities in games that I would not have before.
I have also noticed the rate at which I blunder horribly by hanging pieces and such has definitely gone down.
But I agree that tactics training in and of itself is not the end-all be-all that some people make it out to be.
This is something I don't like about Yusupov's series.. There are so many exercises which are not checked for errors or are left without alternative moves which are just as equal if not better. Here is one of many examples. The move I chose here ( black to move ) was ...Kf5, and once again this is Stockfish's first choice no matter how long I let it analyse the position ( at depth 37 -2.17, second best move -0.96 ) that is a large margin. This is a three star problem and I am about to lose three points even though I objectively found the best move in position.
What are you supposed to play here and why?
That is impossible to answer in one sentence. It really depends on what black would play himself. It could be though, at least from a practical point of view, that Kd5 is a better move after all.