You have less time to think in a Blitz game than in a long game, but so does your opponent. If the time constraint causes you to make more blunders, your opponent makes more blunders as well. So why’s Blitz harder than Rapid at equal Elo ratings?
That's a very good question.
The simplest answer is that ratings are not an absolute measurement like height or weight. Ratings only describe a distance (so to speak) between you and some other guy. For example 200 points. Whether that means 1000 to 1200 or 20000 to 20200 it's the same as far as the math is concerned. Chess.com could, for example, set the average rating for rapid at zero, so people below average will have negative ratings (yes the math works just as well with negative values).
So the difficulty of a time control in terms of rating is a combination of where chess.com has chosen to set the rating range, and then the players that join that rating category. For example if very few high rated players join the 10|0 time control, then it may be impossible to get a rating of 3000 even though some people have a rating over 3000 in blitz and bullet. (Yes, in theory the rating system could put you at a rating of 3000 even if the next highest rated player is 800 points below you, but the effectiveness of the math breaks down when the gaps are around 400 or higher.)
Beginners are often told on the forum to stop playing Blitz/Bullet and to focus on longer (Rapid/Standard) games since longer games allow more time to think. This advice is good, but I don’t understand why Blitz and Bullet actually are more difficult for beginners when it comes to Elo (many players (especially beginners) have Blitz ratings far lower than their Rapid ratings). You have less time to think in a Blitz game than in a long game, but so does your opponent. If the time constraint causes you to make more blunders, your opponent makes more blunders as well. So why’s Blitz harder than Rapid at equal Elo ratings?
My 623 Rapid rating is almost 500 points higher than my terrible 125 Blitz rating. Rapid goes normally for me, but Blitz is a nightmare. In Blitz, I often find myself already losing before openings are over (to players rated under 200!), but this doesn’t happen in Rapid (in which I beat players rated 600+).