They’ll do it anyways, so why not make it less horrible for them?

I disagree with that. Will they do it anyway? There is evidence that putting up simple barriers to suicide (such as guardrails on a bridge) is effective at reducing suicide, while having a method of suicide readily available (such as a gun) can increase risks of suicide. Suicide is often an impulsive and irrational decision.

If some percentage of people would be deterred from suicide by the inconvenience of doing it themselves, and some percentage of that group would go on to recover enough to lead happy lives, wouldn't that at least potentially be a good enough reason to restrict it?

But to answer your previous question, yes. We do let people suffer until society changes. Because I believe that it is better to endure the suffering and injustice caused by society than to look for an easy escape that doesn't actually solve the problem, at least for anyone else. If I see suffering, is the proper solution to rip out my eyes? No. That's incredibly misdirected, but that's the logic of suicide. Rather than seeking to address the actual problem, it's directing violence towards one's own ability to sense and perceive the world around them. It is the ultimate form of "out of sight, out of mind," taking it so far that you eliminate your own mind for having the audacity to report to you about unpleasantness. Addressing the underlying cause is what's important, the pain is merely a symptom, which exists for the reason of telling us something's wrong.

There are exceptions to that generalization. It is possible that the real source of the problem is within one's body, that it's causing incurable and unbearable physical pain. In those cases, I think it's acceptable - but no further.