"Through what evidence do you conclude the inevitable heat death of universes?"
Entropy and the law of conservation. The amount of usable energy goes down over time and can never be increased. I've read The Last Question of course but the reversal of entropy is one of those things which, in my opinion, is necessarily impossible or the physics upon which the rest of this concept is based would be invalidated to begin with.
"Wouldn't the machine race we're about to give birth to qualify as exactly that?"
Sure, but it does not have to be machine life. Literally anything which can self-replicate and survive in space will result in the same outcome. It does not matter whether it's von neumann probes, engineered biology or just humans in space habitats with brain implants. Many roads, one destination.
Incidentally there are good reasons why we might prefer it be engineered biology, or stupid machine replicators rather than deliberately engineered AI. We don't know for sure if we can create truly conscious AI. But we do know that evolution can create authentic consciousness. Allowing machines to evolve to that point on their own is messier and slower but will be more certain to result in an authentically conscious descendant.