"You can't click sleep moves if you have another option." is a cleaner formulation than "You lose if you sleep a second Pokemon." You don't get ugly interactions with Trick/Encore, you aren't tempted to risk a loss to put something back to sleep, and new players can't click a button that makes them insta-lose.
This does rule out PP-stalling or trying to re-sleep something with Spore, but that's a cartridge-inaccurate strategy that gets criticized a lot when Sleep Clause gets discussed. Between getting rid of that strategy and banning sleep altogether, the nerf is the better option if you think sleep should stay.
(I'm personally fine with keeping Sleep Clause Mod as-is, but there are a lot of people who want it changed. Settling the cartridge accuracy debate would be nice, if only so sleep can be judged on its competitive merits and not on the way a clause was implemented as a mod 20 years ago. Sleep Moves Clause is not the only way to restore cartridge accuracy, and whether sleep is broken has nothing to do with whether Sleep Clause is a mod.)
Dire Claw and Relic Song are interesting corner cases, but not impossible to handle. One option is to say that the secondary effect is negligible and allow them to be clicked freely, meaning there will be some cases where you can randomly sleep a second Pokemon. This is analogous to what happens with freeze right now, and we can always ban the moves if they become a problem.
Another option is to lump them in with the primary sleep-causing moves, meaning they can't be clicked if an opposing Pokemon is already asleep. That would be another minor nerf to sleep moves, since the secondary effect moves can no longer be spammed once they proc. The upshot is that this prevents annoying cheese strategies that violate the spirit of the clause, but it also messes with Meloetta's mechanics if Relic Song gets a sleep.
Effect Spore is harder to control, since you can't turn off an ability. But the risk of getting double sleep from Effect Spore in OU is low, and if it ever becomes a viable strategy, it can always be banned.
Natural Cure, which was brought up in the other thread, is actually the hardest case. Can you click a second sleep move if you don't know whether the opposing Blissey is still asleep? To err on the side of caution, I would say a switched-out sleeping Pokemon should only count as awake once it has been seen to wake up.
TL;DR: Sleep Clause does not have to be a mod. Non-mod Sleep Clause does not need a lose condition. The corner cases are not insurmountable.