Repetition: 5.3.1. 1. A player may claim a draw if the same position occurs three times. 2. A game is drawn if the same position occurs five times.
50-move rule: 5.3.2 A player may claim a draw if the last 50 moves have been completed by each player without the movement of any pawn and without any capture.
75-move rule: 9.6.2 The game is drawn if the last 75 moves have been completed by each player without the movement of any pawn and without any capture.
https://www.fide.com/fide/handbook.html?id=208&view=article
This seems like a good functional chance, and it's good to stick with FIDE rules unless there are good reasons not to.
Doesn't the server client automatically terminate games at threefold repetition & 50-move rule.
I have no objection to the current implementation being changed. Currently:
The "auto-claim" user setting could be reworded could be used for auto-claiming the 50-move rule.
The 5-fold repetition has been implemented recently, why not do the same for the 75-move rule? I don't see any cons.
If a position is reached where the 50-move applies, it would then show the possibility to claim draw

I completely agree with @kraktus. The 5-fold repetition implementation is really nice and the 75-move-rule could be implemented just alike. It's wrong to automatically end a game by draw after 50 moves without capture or pawn move.
As an example this game https://www.chessgames.com/perl/chessgame?gid=1959810 between Firouzja and Demchenko could not have been played on Lichess. It ends after exactly 75 moves each without capture or pawn move. Lichess would have ended it after 50 such moves. According to the PGN it was played on chess.com, which I don't know how. On chess.com games for normal users terminate automatically after 50 such moves.
5-fold done via https://github.com/ornicar/scalachess/commit/8ca89d7e61ccf7035f25dd03d5222ccf81e167d7.
Always automatically claiming 50 moves seems reasonable (but impratical OTB) and in the same spirit as the 600 ply limit on Lichess. I've yet to see a game where both sides have a legitimate reason to continue.
I don't look at this the same way. Per the FIDE rules the players are allowed to continue beyond 50 moves without capture and pawn moves. To end the game after 50 moves automatically is a rule which does not exist in FIDE rules. When the players want to continue beyond 50 such moves, they should be allowed, for the FIDE rules allows it.
As such, I don't think this needs an example game "where both sides have a legitimate reason to continue". When the players want to play beyond, let them play. They know what they do. Also, it is confusing when each software implements it's own rules here when there are official rules. Lichess should implement the official FIDE rules, per my opinion.
However, here are three example games, all between GM's and a perfect demonstration that play beyond the mentioned fifty moves can make sense. So now you see three games where both players "have a legitimate reason to continue".
I already gave the example of Firoujza vs Demchenko at the PRO League Group Stage 2019 (https://www.chessgames.com/perl/chessgame?gid=1959810)! They decide to continue beyond move 101... Ra5+, fifty moves without capture and pawn move. Lichess would have terminated the game here. It's a KRPPP vs KRPP endgame. Firoujza has a reason to continue, being a pawn up. Demchenko may need the win and hope for flag-fall or blunder by Firoujza. Just remember, when Firoujza runs out of time, Demchenko wins the game, as there is a theoretical win in this position for Black.
A second example is Onischuk vs Guseinov from World Blitz Championship 2014 (https://www.chessgames.com/perl/chessgame?gid=1760281). The game results in a KBB vs KN endgame. After move 111... Nc6 they played fifty moves each without capture or pawn move. At this point, White can win per tablebase (https://syzygy-tables.info/?fen=2B5/8/1kn5/3KB3/8/8/8/8_w_-_-_0_1)! So White has definitely a reason to continue. As in the first example, the other side can hope for the flag-fall or blunder.
A third example is Cheparinov vs Jones from IMSA World Masters Blitz 2019 (https://www.chessgames.com/perl/chessgame?gid=1959266). This is a KR vs KRN endgame. Here Lichess would terminate the game after 118... Ra6. Both players continue, one might hope for the flag fall or blunder, the other for the win. This is extremely difficult to play, full of nasty traps and to continue in a Blitz makes perfect sense. They are not machines; they are human.
So if the players want to play, then let them play. They are allowed by the rules to do so, and these games also show that you would influence the game in this way. The games ended in draws, but that was by no way clear. Lichess would have ended all these games before the player wanted to do so. I think this would be sad, for when the players want to go for the win, let them play, it is exciting also for the viewers. I am sure, on lower levels blunders happen a lot, so such games also do not end in draws. Here ending the game automatically after fifty moves without capture or pawn move would even "falsify" the result.