The clock always matters in real time chess.
The limit where the clock matters more than the actual position is, from my experience only, about 30s + 0 HyperBullet. Less than that, 15s + 0 UltraBullet, is closer to 2D aim...
For me, 2+1/3+0 is just too fast to play chess. I think mostly because I started relative late(in my 20's) and I don't have the raw instinct. At one point my brain just breaks, often at one point I just can't stop myself to take some time to find the best/good move and I lose on time. Chess improvement wise both time controls are for me to fast, but for both is something to say to get more calm under extreme time pressure and just time management in general(since I even lose regularly in for example 15+10 on time).
On the other hand 5+3, I can reasonably play a game of chess. So despite being still weaker than my rapid/classical it is not out of this world. 2100 is well within reach by this point. So basically playing 3+0 makes no sense for me. I will just drop down a lot in rating(easy to 1700s or even lower). Personally, I don't care that much, but if I then again start playing 5+3 would be just no fun for my opponents. Even in the 2000's my level is usually already well ahead of my opponents, but my time skills are way less :p )
SO then 2+1 is ideal for me. To just 100% focus on getting a bit faster without tanking my rating for the 5+3 pool.
Extra note: for me, 3+0 is still bullet. And 5+5 is blitz(not rapid). 1+0 is already ultra bullet :p
I agree that 5|5 is Blitz (or should be Blitz), yet another "Rapid hole" to pointlessly argue about. :-D
