In all honesty, how can people in Bullet play so extremely fast?

I can address these things for you too

The first part you wrote is true that bullet players do need to be more than just fast; it helps to be able to calculate quickly and fairly accurately. They don't necessarily need to find the best moves quickly, but they do need to be able to find good moves quickly. Where I think you begin to slip in assumption is that these bullet players (especially around 1000 rating) must know opening theory to play this quickly. This is not necessarily the case. Many of these players focus more on bullet/blitz than rapid and so it makes sense they have experience in this area. They have developed (mostly through trial and error) a general understanding of their chosen opening, but not necessarily a deep understanding on opening theory, or how to exploit opening errors by you. These players have learned (consciously, or subconsciously) to simply stick to their setups in positions comfortable to them and to follow opening principles when they aren't as sure of the continuation. This may not be optimal, but it does tend to avoid most trouble and this is what they want because bullet doesn't have much time to spend on calculating a precise way out of a troublesome situation they've never seen before.

As for your main question of how a bullet player who is rated 700-900 rapid can beat you at bullet (and sometimes easily as you describe), this is because you aren't playing rapid. They are better adapted to the time control and context of blitz/bullet than you are. That's where their edge comes from.

Your observation is also true for rapid and surprised me as well. I'll be playing a rated rapid game with an opponent close to my rating and I'll occasionally notice their blitz/bullet rating is rated about 300-400 points higher than their rapid. That's 300-400 points of higher ability AND they get more time? Shouldn't I get cooked? Surprisingly no; I've won several of those games.

In that example, what typically happens is they play a good (although not best) move and then I'm able to exploit that because I have more time to calculate and find the "best" counter to their sub-optimal move. Having extra time doesn't necessarily make you higher rated because the extra time your opponent gets as well may be the thing they need to find the way to breakthrough your chess position.

If there's an untold "secret" to playing bullet chess we haven't mentioned yet, then it is probably this ability to play good moves quickly, but not necessarily the best moves. They are able to do this mostly through experience with speed chess (lots of playing), mostly following opening principles (control the center, develop pieces, castle early), and through decent pattern recognition (like recognizing opening positions that they are familiar with, so they don't need much time to play a move on autopilot).

What I described above also comes with a weakness though; this is how rapid players typically exploit players with lots of speed chess experience: the weakness is this autopilot speed they've conditioned themselved into. You outperform this by putting them into positions where they are forced out of this autopilot mode, or by having the attention to detail to find the "best move" better than their "good move", but this typically requires calculation ability and more time on the clock, so rapid is better suited for this.

Different timers = different skill sets.

I don't worry about what my opponents are rated in different timers. If you're playing against them in bullet, then their bullet rating is the only rating to consider.