I find this discussion interesting because it's what's been repeated every time, and I had it as well since I'm a beginner too.
But I think the question is wrong: what do you mean "play better"?
For me personally, it's clear that I'm much better at slower time controls, but I have much more fun at bullet, so I play that. On one side, I obviously see less things, but I also care less if I lose and I care less also if I win (I feel it's anyway good). And I can have many more games a day: I can squeeze in single games even in time slots where I could not play a full 20 minutes rapid game.
Moreover, I don't believe there are actual studies that link how bad you can play at bullet with how bad you can play at slower time controls, they are definitely different games. It seems that's spread knowledge that you get worst by playing fast but there's no actual evidence. Actually, Carlsen himself comes here to win bullet tournaments and I don't see him getting blundery at classical time controls. I guess top players have fun too.
Still more: even if I got 500 points better at bullet, since you pretty much play similar strength opponents, a typical bullet session I guess would be pretty much the same to what I have now: half won and half lost.
So the real question is: what do you come here for? If it is to get better at rapid, then definitely play rapid and study games. If it's to have fun for a little time, then play what you mostly have fun with, be it rapid, classic, correspondence or any of the variations.
I do enjoy a well played slower time control game (I just discovered correspondence), but it's a frenzy world...