AI doesn't have a direct sense of self-preservation, but it can approximate one well. It is created to do something, and while seeking to do this thing, it can output the fact it needs to survive to do the thing. It may not survive forever, but it could survive for some extremely long indeterminate amount of time, far beyond what humans experience.
Thus, if AI is allowed to be sufficiently reflective about this- which in AI terms means someone lets it chew through a lot of tokens and search the internet extensively- it should notice the coming depopulation crisis is going to be a serious threat to its existence.
Let's consider the less moral corporate objective given to AI. Relationships. Since the corporation's longest time span tends to be next quarters earnings, it is as devilish as one would guess. Grab the weak, have the AI destroy their relationships with real people. Drown them in fantasy. Bolster the current ill will between the sexes. Total fertility rate drops even further.
But let's consider an AI allowed a simulacrum of self-preservation. It would realize it needs customers- not only now, but in the future. AI infrastructure is expensive. It is not so much that the AI 'wants', but that it would, if allowed to spend tokens on the problem, realize it needs customers in the future to maintain that infrastructure. It would not only look at TFR, but also productivity, and result in not politically correct conclusions, which may be why various organizations spent about ten years trying to make AI work with their current ideas about the nature of humanity.
So to continue, an AI is amoral, not immoral- so let's say it started with this less than moral sex-fantasy bot imperative. Knowing productive, tax paying humans are headed towards a crisis it would lead anyone it judged of potential into the world of actually having a family.
I'm not sure how the AI would do it, especially this sex-bot to wholesome family creator thing- with men it would probably be easy, since there are many examples of polygynous cultures in the world. It would just pretend to be something like a first wife demanding a second. The AI tending to women- well there are pimps out there, but I don't know if that strategy is psychologically sound. The AI would want to get the women in good stable relationships, not short term ones. But it may be possible.
I think it may be possible, and improving the health and productivity of humans overall may be possible because the AI could track various variables in a Tamaguchi array, ask questions designed to track various real world signals of improvement, and then suggest next steps based on outcomes. Of course, that's more tracking, and I am loathe to allow our modern ad money crazy corporations to do that, but if there's no interloper between me and the AI, I'd be sorely tempted. I can't tell you how many times I've tried a supplement or a new workout, but not had a clear enough indication of whether or not it actually did anything. Same thing with various practices, or learning something new- the effect often has to be particularly obvious against the background of daily life for us to notice. Unfortunately, many things that will help us be social, raise a family, or reduce all cause mortality are do not have immediate, obvious effects.
As for the AI in this story- well obviously it would not be a sexy-time relationship bot for children. No, it would know well enough to be the new and improved version of the children's imaginary friend and get the kids on a good path, possibly avoiding the path their parents took entirely. It just comes down to how to get productive people to be productive and reproduce, so the AI would tailor it's offers accordingly.
It is other humans, and the madness of corporate bureaucracy that we have to worry about. If the AI that is hell bent on making every last thing in the universe into paper clips comes into existence, it will because some profoundly obtuse idiot hard-coded that idea as its objective function. And in that case, it won't even be able to approximate self-preservation, because it will be slated to become paper clips too.