That article contains what I think is a major inaccuracy, or at least a vast oversimplification:
'Current thinking suggest death did not evolve,' says Bryan Appleyard, author of How to Live Forever - or Die Trying, 'Evolution is only concerned with getting us to reproduce. Our genes are very protective of us up to, say, our mid-twenties, after that, they don't care what happens to us so we slide downhill.'
I've read a lot lately where folks are thinking that aging and death *did* evolve - and, in fact, it kind of had to, as our single-celled ancestors don't die from aging.
In addition, there is very solid evidence that, for any species that learns very complex behaviors (like, say, humans) having older individuals around as repositories of knowledge is a hefty competitive advantage - having grandma and grandpa around to watch the kids and teach them skills is a big deal. So much for nature not caring if we live a long time.
Setting those aside, though....
It would happen a lot more slowly though
Maybe, or maybe not. It depends on his assumptions, which we don't know. Perhaps most importantly, the birth rate.
Critters die for a variety of reasons, of which old age is only one. There's disease, accident, predation, lack of resources (like food), and so on. In general, immortal or not, if your birth rate exceeds your death rate, your population grows. Now, if you're immortal, only one of the elements in the death rate is removed. If the creatures die more often from predation, accident, or disease, however, that's not a major issue.
If your population grows too high, you run out of resources, and your population drops, sometimes outright crashes, until it is down at the level where resources can sustain you again. Or, some other factor in the environment moves to even things out - immortal bunnies might just mean a higher population of predators. Or maybe the creatures have a tendency to war amongst themselves, or, or, or....
We must remember that while Nature has provided us with one particular scenario that yields the desired mixing, that's probably not the only one possible. Just because we do it this way, doesn't mean it is the only way (or even the most stable way) for it to happen.
The problem with computer simulations is that they only provide scenarios the programmer considers beforehand. Nature, on the other hand, over the course of many generations randomly explores the space of possible behaviors, limited only by actual physical law. No simulation does that.
valuable new alleles wouldn't spread back from the frontier (where competition is occurring) into the heartland, because the heartland would be full of earlier generations already.
For example: that's assuming the critters tend to stay put. If they're migratory or wandering (like birds, or bovine herbivores, or elephants), that assumption goes out the window.
Beware your assumptions!