It gives the TD some wiggle room to prevent what someone is trying to pull in the OP which is to (in effect) use the opponent’s scoresheet to validate a claim. The TD could allow the player to borrow the scoresheet on their own time provided that the updated scoresheet not be used for any claim on this turn.
Borrowing has to be on your own time in this scenario per 15D2:
15D2. Borrower’s clock runs.
The clock of the player making such a request is running and shall continue to run until the scoresheet has been returned.
In the original post it’s unclear whether the borrowing request comes before or after the claim of triple repetition - if the clock is still running I don’t particularly see the issue.
Separately, denying an accurate triple repetition claim seems like a bit of a jerk move.
Isn’t that what I said?

Isn’t that what I said?
It’s part of what you said:

The TD could allow the player to borrow the scoresheet on their own time provided that the updated scoresheet not be used for any claim on this turn.
But it’s unclear why the updated scoresheet could not be used for any claim on this turn if the borrowing request was made prior to the draw claim, which the original post does not explain.
If Player A (the one planning/making a draw claim) has an inadequate scoresheet because he has stopped keeping score by being under five minutes, then he runs afoul of
15D1. Clock times.
Both players have at least five minutes remaining in the current time control.
so he does not have (in effect) a right to borrow Player B’s scoresheet even on A’s time (which I assume is there for precisely the reason we are discussing—absent the need for a claim, I can’t see any reason to deny someone the option of using their dwindling time to update their scoresheet).
Now if A has a position where there is no way for B to make progress, so either a triple occurrence or a 50 move claim is inevitable, then A has a number of remedies, such as restarting the score, asking for a TD to observe repeated moves or getting help counting moves. But if B basically blundered into what may be a triple occurrence, then I’m not sure which is jerkier—B denying the triple occurrence (it is, after all, A’s obligation under the rules to support it), or A having availed himself of the right to not keep score, knowing that it could make it impossible to validate a triple occurrence claim, asking B to (in effect) validate it for him.
However, yes, I am not thrilled as a TD to have to verify a breathtakingly obvious triple occurrence claim.