It doesn't. But if FTL is possible, ASI will get us there faster.
> How do you support a population for 400 years
There are engineering solution to all of the individual points you're making.
> Basically, what you're arguing is that exponential growth in AI (not a given) will unlock huge material sciences and biological advances.
It already is unlocking material science and biological advances. What happens when you can literally simulate matter, the human body, cells, perfectly in a computer - and run millions of experiments to reverse engineer these things? I don't understand how someone can't see where this is going.
> While possible, it's unlikely to combine in a way that suggests "we should send people to other solar systems".
I'm not sure what your last sentence means exactly. What determines this is whether people have the desire to or not, and I'm sure profit will be involved in one way or another. If there's a will, there's a way - there are enough sci-fi movies and books to clearly illustrate what mankind dreams of and has dreamt of for millennia.
Just my thoughts as a software engineer with a mechanical engineering background.