Why is it allowed to request an increment in time trouble? From the FIDE Handbook:

III.2.2 These Guidelines shall apply only to standard chess and rapid chess games without increment and not to blitz games.
III.4 If the player having the move has less than two minutes left on his clock, he may request that an increment extra five seconds be introduced for both players. This constitutes the offer of a draw. If the offer refused, and the arbiter agrees to the request, the clocks shall then be set with the extra time; the opponent shall be awarded two extra minutes and the game shall continue.

According to this, if I crush my opponent in a rapid game using all of my clock time except for a minute, then I can simply request an increment of 5 seconds for continuing the game (my opponent gets two minutes, but that will not affect the game due to his/her hopeless position). Is this fair enough?
There have been many examples where this rule could be used but wasn't, for instance, the Unofficial Chess960 World Championship, Carlsen could have requested an increment and tried to win the R + B vs R endgame instead of flagging. Why didn't he do so?
but that will not affect the game due to his/her hopeless position


Well, if your opponent has a hopeless position, they would be better off accepting the 'automatic' draw offer which is implied whenever you make this request: "This constitutes the offer of a draw."

Therefore, invoking this III.4 rule is only beneficial if you have a worse position and very little time left on your clock, which sort-of makes sense: especially in classical games, we want the stronger side to win on the board, not by simply moving their pieces around.

One of the strengths of this rule is that it's an objective rule. An older rule (III.5.1) is that the player in time trouble can claim their opponent isn't trying to win by normal means; validating this claim required a subjective decision by an arbiter. I had never heard of rule III.4; I wouldn't be surprised if once it's wider known, rule III.5.1 will be abandoned entirely.