This reminds me of doomsday argument, and even more of  "black balls" from Bostrom's Vulnerable World Hypothesis.

But I find some issues with it:

First of all there's no guarantee that there is such a technology that inevitably leads to extinction. 

Second, even if there is such a technology, there's no guarantee that we will ever develop it, regardless of how many people are born in the future. (We might use proper safety measures and avoid it forever, or until we go extinct from some other cause)

Third, even if we do develop it eventually, the speed of its arrival probably depends on many of factors of which the total number of human lives probably isn't most important. (As more important factors I'd mention, presence/absence of AGI/ASI and whether they are aligned, whether we are pursuing differential technological development or not, how robust our institutions are at preventing existential risks, how good is global coordination and cooperation, how closely the development of potentially harmful technologies is monitored, etc.)

Fourth, even if such a deterministic relationship does exist, and 200 billion human lives inevitably leads to development of such a technology, from utilitarian point of view it doesn't seem to matter much when we'll reach 200 billion humans who have ever lived, as whenever we reach it, the total amount of humans who have ever lived will be the same.

In my opinion the trajectory of population might be more important than the sheer number you end up with.

Ideal trajectory is that of slow/moderate growth as long as the planets we occupy (Earth and whatever other planet we occupy in the future) can support increasing number of people without significantly compromising living standards and exhausting resources.

After that point population should stabilize. Neither grow nor shrink but stay constant.

Ideally population should stop growing at around 80% of the carrying capacity of the planets we occupy, as approaching 100% would be too risky.

Now population decline is typically a bad trajectory for 2 reasons: 

1. It is a symptom of some problems in society (unless intentional)
2. Over the long term it leads to extinction

For this reason, I think population decline can only be good as a short term intentional / consensual process in overcrowded worlds.

If we're already overcrowded (which I doubt), in that case it wouldn't be too bad if population falls a bit, but it must be a short term and reversible decline.

And we're witnessing some concerning trends, like TFR staying well below replacement for long term and failing to recover in most of the developed nations. And the developing nations are also on the same track, just lagging a couple of decades.

Even if population decline improves sustainability, that would be a good consequence, but the process itself might still be bad, because it's not result of intentional decision, but of increasing number of people failing to reproduce in sufficient number  due to various limitations of our society. So population decline, even if it improves sustainability (which would be a beneficial side effect), would still reveal some structural problems of our current society and economic system, and these problems would likely still need some fixing even if we decide it's OK for population to fall a bit for a short while.

The point is that we should be able to reproduce in sufficient number if we want to. It seems that right now we aren't even able to do it. Most of the people end up with smaller number of kids than what they wanted.

Depopulation is Bad

Since I believe that humans have positive both instrumental and intrinsic value, I believe that depopulation in principle is bad, because if it's an entrenched long term trend, it would lead towards extinction. That being said, short term population decline is not necessarily the worst thing in the world, and could even be beneficial in certain extreme scenarios, such as severely overpopulated and resource depleted world, which I don't believe to be the case for our world.

Regarding instrumental value - I think we have a potential to have an extremely positive instrumental value in the Universe. If things go in a proper direction we could populate the Universe with sentient beings who live in utopian conditions. We could create digital worlds also filled with beings who live in delightful conditions. We definitely have potential for this, and I believe we have enough wisdom to actually achieve it. Not achieving this would be, as Bostrom points out, an astronomical waste. Right now we're the only known species with such potential.

On the other hand we're also causing a lot of farm animal suffering (but perhaps we're also reducing wild animal suffering even more), and there's a risk that we could export suffering to other worlds in the Universe, which would be astronomically bad.

But none of this is given or guaranteed. It depends on us. We have agency. Since we have agency, we can use this agency to produce enormous amounts of positive value both for ourselves and other sentient beings. So, I root for us, and I believe that we can and we should fulfill our potential for doing good.

Regarding intrinsic value - I also believe we have positive intrinsic value. Most of people are glad that they are alive, and they always seek to prolong their life if they can. That's the default attitude of most of the people towards life. Not only is life filled with many pleasures (food, music, art, sunsets, reading, philosophy, friendship, love, sex, etc...) but we find many aspects of it meaningful. Even our struggles and challenges can sometimes be seen as meaningful, such as in case of mountain climbing or marathon running. This, of course, is not to say that suffering is good - the types of suffering that can be "good" and meaningful are very rare and very specific, and generally mild. (Very few people would consider marathon running to be torture)

Now to go back to depopulation itself. By default it's bad. But it's not only bad because it reduces population, but it's also bad because it's a symptom of dysfunctional society in which people struggle to fulfill one of their most basic biological functions, to reproduce. There must be something deeply wrong about society in which people struggle to reproduce. Sub-replacement fertility is one of the symptoms of such social pathology.

In my opinion, for a society to be considered healthy, it must have at least stable population. Growth isn't the prerequisite for health of society, but stable population is.

The only exception to that rule would be an extremely overcrowded world in which people intentionally and consensually  try to reduce their numbers by procreating less, until they reach a sustainable population. But that's an exception.

As a rule, depopulation is pathological, especially if it's unintentional, which I believe is the case in our world.

I also believe that recognizing depopulation as a symptom of social pathology could be a starting point in our efforts to improve our societies - not only in order to bring TFR above 2.1 again, but also for its own sake (as everyone prefers to live in a healthier society)