Somebody claims a draw by repetition, the arbiter comes over and asks for both scoresheets. If yours stops at move 24 and the claim is about the positions after moves 31, 35 and 39, you have no way of proving anything. That is the real reason the rule exists, and it is why arbiters take it seriously.
What Article 8 actually asks for
Each player records their own moves and their opponent’s, one after the other, as clearly and legibly as they can, in algebraic notation, on the scoresheet the event provides (Art. 8.1.1). Both players, both sides’ moves, all game.
You are allowed to reply before you write, which is what everyone does when the position is sharp, but you must record your previous move before you make another one (Art. 8.1.3). The scoresheet stays visible to the arbiter throughout (Art. 8.2), so keep it flat on the table rather than in your lap.
Draw offers go on it too, marked with the symbol (=) by both players. And at the end of the game both players sign both scoresheets with the result (Art. 8.7), which is the moment a resignation or a draw becomes official paperwork rather than a conversation.
If you cannot read your own handwriting at move 40 you have not kept a scoresheet, you have kept a doodle. Write the moves the way notation is meant to be written, and put the clock times in the margin every ten moves.
The five minute let-off
Art. 8.4 lets you stop. If you have fewer than five minutes left in a period and you do not have an increment of thirty seconds or more per move, you are excused from recording for the rest of that period. Then, as soon as a flag falls, you complete the scoresheet before moving another piece.
Notice the condition. On a 90+30 control that let-off never arrives, because the thirty second increment is there from move one, so you keep writing to the end of the game. On an old style control with no increment, the last ten moves of a time scramble are usually unrecorded on both sides.
Rapid and blitz do not require a scoresheet at all (Appendix A.2), and you keep your right to make claims regardless. You can ask the arbiter for a sheet if you want to record a rapid game for your own use.
It is the evidence for every claim
A threefold repetition claim under Art. 9.2 and a fifty move claim under Art. 9.3 are both checked from the scoresheets. Get one wrong and your opponent receives two extra minutes under Art. 9.5.3, and you cannot withdraw the claim once it is made.
So the player who counts moves in the margin during a long rook ending is not being fussy. Fifty moves is a long way to count from memory, and the ending where it bites most often is a rook endgame where one side is grinding away with a small edge.
The selfish reason to keep a clean sheet is better than all of that. Your scoresheet is the only copy of the game that exists. Type it in the same evening, find the move where you went from better to worse, and you have turned four hours of chess into a training file.