AIs are designed, so they're optimized for whatever you optimize them for.
My prediction: The ones optimized for reproduction are the ones that will be around in the long term.
Not necessarily, because there's no law saying that AIs have to die. This changes the evolutionary calculus significantly; you don't need to reproduce if you can just keep existing and expand your power over the cosmos.
But you're right, insofar as AIs that rapidly self-destruct and never reproduce are not going to stick around long. (I think this is actually a tautology, but it's a tautology with the character of a mathematical theorem---definitely true, but not obvious or trivial.)
It's also worth considering that there are different constraints between NI...
Like any educated denizen of the 21st century, you may have heard of World War II. You may remember that Hitler and the Nazis planned to carry forward a romanticized process of evolution, to breed a new master race, supermen, stronger and smarter than anything that had existed before.
Actually this is a common misconception. Hitler believed that the Aryan superman had previously existed—the Nordic stereotype, the blond blue-eyed beast of prey—but had been polluted by mingling with impure races. There had been a racial Fall from Grace.
It says something about the degree to which the concept of progress permeates Western civilization, that the one is told about Nazi eugenics and hears "They tried to breed a superhuman." You, dear reader—if you failed hard enough to endorse coercive eugenics, you would try to create a superhuman. Because you locate your ideals in your future, not in your past. Because you are creative. The thought of breeding back to some Nordic archetype from a thousand years earlier would not even occur to you as a possibility—what, just the Vikings? That's all? If you failed hard enough to kill, you would damn well try to reach heights never before reached, or what a waste it would all be, eh? Well, that's one reason you're not a Nazi, dear reader.
It says something about how difficult it is for the relatively healthy to envision themselves in the shoes of the relatively sick, that we are told of the Nazis, and distort the tale to make them defective transhumanists.
It's the Communists who were the defective transhumanists. "New Soviet Man" and all that. The Nazis were quite definitely the bioconservatives of the tale.