>At that point, the birth rate would quickly rise again, no?

Hospitals failing. Caloric intake failing. Yeh sure, why not? Let's just imagine that all high technology that acutally props up reproductive success seriously strained or even vanishing will result in things magically correcting.

The same attitudes that have negatively impacted fertility rates don't disappear because things get worse. They're reinforced by it. An economy that collapses because there aren't enough workers doesn't make people say "I want to have babies"... if the media is to be believed, people wait until the economy improves first. Why would that happen?

>So it's likely that either there's a point of equilibrium

All evidence to the contrary? We've actually run experiments with animals. After they crash, the stress and trauma imposed on them keeps them from reproducing at a behavioral level despite resources being abundant.

>Total extinction seems unlikely.

Based on what exactly? Your unwillingness to own up to plainly obvious but disturbing conclusions?

>Because if you're back to living in a pre-industrial society,

Short of time travel, we can't have a pre industrial society, as that's the society that comes at a point in history before industry. What we'd have would be a post-collapse society. They're not the same thing. They don't even resemble each other much.

When moose on the island are nearly wiped out and their predators starve to death, the moose can "swing back up" because moose never have low fertility. It's something like 25.0, give or take. They have a low population not because fertility went low... they have it because baby moose get eaten or starve or die from disease. Once those pressures ease up, the 25.0 fertility is still there and population rises quite rapidly.

When human fertility goes down, it means that humans can't ever bounce back. So unless you're hypothesizing some sort of thing which causes human fertility to rise paradoxically and inexplicably, this reply of yours makes no sense at all. Not even a little. By the time people like you realize how wrong you are, your species will be past the point of no return.