if you like science fiction, you may enjoy reading the bobiverse by dennis e. taylor. it describes exactly that scenario, except that the AI is an uploaded human. but that's pretty much the same thing.

Because hardware never breaks, especially not on galactic timescales, and without resources to perpetually replace failing components.

They will have whole galaxies of resources! Massive amounts of redundancy. Mines, Refineries, and Factories on planetary scales.

Not going to happen tomorrow, but perhaps in the next few thousand years something will be ready to begin its journey.

I'm in an AI cult. Send help. No don't.