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Given the existence of the (optional) 50-move rule and the (mandatory) 75-move rule, are the (optional) threefold-repetition rule and the (mandatory) fivefold-repetition strictly necessary? Do the repetition rules serve any purpose other than to (potentially) shorten the time to draw of a game that is futile for both sides?

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An important, but not the sole, reason for allowing draws in chess (outside of stalemate) is to prevent infinitely long games. Hence, when helpmate is unachievable for either side (a condition loosely termed "insufficient material"), a draw is immediately declared.

Now, n-fold repetition is introduced so that draws can be claimed in positions where neither player is willing to tolerate the risk involved in deviating. Hence even if helpmate is available (indeed, even if checkmate is achievable, but beyond the player's search horizon) mutual consensus may be that a draw is safe. Note that n-fold repetition is by itself sufficient to ensure no game is infinitely long, since if the number of moves reaches n multiplied by the total number of possible chess positions, it is guaranteed that at least one position has been repeated >= n times. I know of no estimate to suggest how long this would take with, for example, random movers or in the worst case, but I imagine it would take astronomically many moves.

To come to your question: does the N-move rule (N moves without piece capture or pawn move) stand on its own as well? The answer is of course that it does. It also guarantees termination of a game much more quickly than the n-fold repetition rule does (estimates come in at 'just' 6000 moves for N=50 and 9000 moves for N=75).

So why does the n-fold repetition rule exist? Namely, as implied earlier, it serves a distinctly different purpose from the helpmate-impossible and N-move draw rules. In particular, the latter rules are implemented to prevent an infinite-length game when it is clear that either any termination is impossible (helpmate-impossible rule) or when play is seen not to head towards any termination besides a helpmate-impossible draw (N-move draw rule). The n-fold repetition draw rule exists primarily not because it provides the above theoretical bound on game length, but practically, because it allows players to limit their risk or cost in scenarios where neither find it worthwhile to deviate (whether it decreases the ground-truth positional evaluation to do so, or not).

Under such circumstances, if neither player feels sufficient incentive to play anything other than the repeated response, requiring 50 repetitions of the same moves seems like a waste. Better to call it a day early.

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are the (optional) threefold-repetition rule and the (mandatory) fivefold-repetition strictly necessary?

Yes, they are necessary. Something which players sometimes forget is that arbiters are people too. We have homes to go to with warm beds waiting. If both players refuse to agree a draw but also refuse to make an effort to win and start repeating the position then after the fifth repetition we can call an end to the game and go home.

You don't think there are woodpushers out there who would do this? Check out this question and answer regarding the 75 move rule.

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