If you want to argue bullet isn't beneficial, you could make the same arguement that tactics trainer doesnt't help because it gives you positions that wouldn't occur in real games.
Well again, the skills you train are the skills you strengthen.
Bullet chess does help with finding OK candidate moves fast, and spotting all the elementary threats fast. If you can do this with some confidence, then you'll be able to manage time scrambles better.
Tactic puzzles help not because they happen to mimic real games (almost all puzzles are taken from real games), they help because you're training skills used at every phase of long games. Visualization is IMO the primary one. The timed tactic puzzles on chess.com focus on pattern building which is useful too.
You may argue that blitz exposes you to patterns too, but they're not very good quality. There may be a fundamental flaw in a tactic or idea that is never brought out over the board in blitz, and when you keep playing it (or perhaps you guard against this non-threat repeatedly) you're reinforcing this bad idea/pattern.
But also, you're not pushing yourself in blitz... you really are able to see more and punish more, but because an advantage on the clock can so quickly trump an advantage on the board you're forced to make superficial moves. If you seriously analyze your blitz games in an attempt to learn, you'll mostly find mistakes that you would have avoided if you'd had more time in the first place.
So to sum up, you're practicing the sacrifice of accuracy for time, many moves you think are good can be bad habits in hiding and the analysis to learn from your mistakes is of very low value.
Furthermore, I dont understand what all the fuss is about. There is NO LUCK in bullett chess -- it is pure skill, just like standard time control (2.5 hours for 40 moves) chess.
You win a bullet chess game by making the best possible move based on the position. In bullet chess you just do it faster. If you can't pick out the best possible move in the position fast enough to compete in bullet chess, then I feel sorry for ya.
As Ivan Drago told Apollo Creed right before the exhibition match in the fine film Rocky IV (1985) . . . "You will lose."A big part of bullet is literally trying to guess your opponent's move. Whether or not you are correct is sometimes largely based on chance. Isn't this luck?