These players who are claiming their grading have shot up in blitz play are no doubt talking about blitz gradings. Blitz gradings I think are not the same as normal play gradings. I have two gradings and my blitz grading is very much lower than my "over the board" time control e.g. about 30 moves to the hour grading. These are actual grades recognised by the chess authorities. If I played more blitz chess my blitz grading might increase, but my normal time control grading would not change because I was not playing games in that time control it would at least stay the same, but I know intuitively that it would actually go down,in the case of then returning to the long game. The reason for this is that most games that are lost are lost because of blunders which are more likely exploited by opponents in long time controls. Playing quicker encourages blunders,because fewer combinations are considered, or analysed properly, which another poster on this site has already stated. Going from quick play back to long play then carries over the propensity to move quicker and carrying the blunders back into the long game. If you think you are wasting your time by taking longer over moves in chess, might I suggest you are wasting you time playing chess at all!
Obviously, blitz ratings are generally lower than rapid because players have less time per move which means there is higher chance to blunder in blitz. Yet, unlike bullet, you cant play only for time in blitz. So, you need a relatively good chess with less blunders in time pressure. And precisely for this reason, it is the ideal format for beginners to improve.
And it is difficult to improve in any format of chess without improving in chess overall. The only exception is perhaps bullet or even 3 min blitz. In those formats, you can get away with playing for time. But, to survive in any format longer than 3 min per side, you need to play relatively good and blunderless chess.
Bullet games are so addictive, but the quality is so much poorer. I had a go again yesterday and I only tended to lose to those who played proper lines, those who played aggressive openings with the queen I was able to pick off, which I never used to be able to do, I'm probably better at bullet after all the daily games I've played. I doubt it works on the opposite way though. Bullet encourages a mentality of "hope he falls for this trick" which slower chess doesn't really need.