Bullet is chess... but so different that the skills to implement differ highly. This difference affects the spirit of the game.
A chess player have a vey close level in different official format of time control : 2h 40 moves 1h KO, 1h30 40 moves + 30 minutes with increment of 30 s, 2h KO.
In this different format, the pressure of time slightlty varies but the skill of a player remains decisive.
To give a simple example, a player without a theoric base (of opening and positionnal evaluation) will rarely get results in competitive (official) games.
Sadly, bullet allows snaky strategy (check chaining without sense, move with tactic error but which requires a long analysis time to see it,...). We can easily found games with a terrible positional disadvantage (around -4) reversed by loss in time or by amazing turnaround. It's obvious this reversed situation don't exist (or anecdotally) in classic format. However, I like play bullet for the pleasure and as a side of the training.
Finally - and i'm sorry for being a little hard in my words - I think the condition to assert what is chess is know chess in official games or in official structure (club, competition) and not only like chess like a game without know how is organized federations.
I agree, one family and time controls do not change whichever version of chess is being played.
Time controls however do affect ratings differently which is something that should be considered.
thanks