The optimal results using any film will be at ~box speed with standard development. I say ~box speed because even the box speed might be slightly off for a batch. In critical film work, one often buys film in bricks or batches so that a roll can be shot at varying ISO's, processed, and then the best ISO chosen for the rest (best being determined by film density). It's not uncommon to change the rating by 1/2 stop.

Pushing does come with tradeoffs and it would be silly to think otherwise. But, people like to be able to "get away" with things. If you can push a film +1 with only a very slight degradation in range and grain, then you've found a way to get faster shutter speeds when you need them.

Also, films react differently to the process, allowing one to experiment with "looks." As xiota notes in the comment, there are so many possible combinations that people discuss it at great length. The same is true on the opposite end if you search for Stand Development.

In my own experience in pushing 135 Ilford Delta 3200 to 1600, 3200, 6400, and 12800 and developing in DD-X...it looks like absolute garbage past 3200. I'm really not even a fan of it at 1600. But, shot in 120 at ISO 800 and souped in Rodinal...that I like. I mention this just to show that no, not all pushing gives desirable results. Also, if a lab is touting it, it's just to make more money off of you as they typically charge more for the process. If a film vendor touts it, take it with a grain of salt. Delta 3200 actual box speed is around 1200 but it's "meant to be pushed." I have found the results dissatisfying pushing to the "box speed" and using the recommended developer. YMMV.