The problem is rates never stay the same. What you say is correct, if nothing changes. But things do change (for one, population is decreasing).

Let me put it another way, if a pandemic would reduce the world population by 50%, what do you think birth rates would do?

Now, if your argument is "it will soon be a shitty time to be old and it will last for a while", with that I do fully agree. The old people of today have it much easier than the old people in 30 years. But people in 80 years might have enough (resources, space, etc.) to start making children again at different rates.

I would say we will live and see, but doubt it is the case for that long periods...

Not that long ago it was believed that fertility rates could never fall as low as they are currently falling because it was thought that people would start having kids again at a sufficiently low fertility. That wasn't just a lay view either, but was part of all population models. It's why the UN had to recently make major revisions to their population models. It turns out that people didn't just start organically having children again because some stat fell below some level. Go figure. My framing is snarky, but is this really a surprise if we think about it objectively?

All of the claims about causal effects on fertility from education, income, and so forth are nonsense. They have endless major counter-examples, the least of which not being the past where they completely fall apart. The cause is solely and exclusively cultural. And completely changing the trajectory that a culture has been headed down for many decades, arguably centuries depending on exactly which component(s) of culture one thinks is responsible, is only viable with very extreme actions. Would a "real" deadly pandemic shift culture? I imagine probably, but even that's still almost more of a hope than a guarantee - akin to population models of the past.

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Finally, the population effects of declines in fertility only really become visible once the first low fertility generation starts significantly dying off, so about 60 years from the advent. This is why Japan looks so terrible relative to other countries, even though their fertility rate is significantly higher than many other places. Japan fell below replacement around the 70s while South Korea lasted all the way til the 80s.

The point I make with this is that the future isn't speculative. South Korea's fertility rate has been around 1 (and even less) for about 20 years, the approximate length of the fertility window for a generation. So they're going to pay a huge price for that in the future, and that is now inescapable. It doesn't matter much if a bunch of 40+ women decide they should've had children - it's now simply too late.

Even if a later generation decides to try to 'fix' the past, the burden is unrealistically high to manage. To get back to the initial population equilibrium, they'd need their next generation to have a fertility rate of 4. A generation at a fertility rate of 1 cuts the generation size in half, so a generation at 4 would double it and return to the initial equilibrium. That's wholly unrealistic.