For a time, but not forever. There are too many benefits to having immortal workers, too many drawbacks to having new people outnumbering the old, and too much danger from long term exploitation. They'll be motivated to find the most stable system in the long run, and eventually they'll realize that exploiting people is dangerous on large time scales.

The rich people of today create a worse world for the children that inherent their wealth at best, or for themselves at worst. Capitalism creates suffering, which then creates instability, which then creates conflict, either with the ruling class directly or with other nations. They might support fascism because they think it's better for them than socialism, but fascists create conflict with each other. The devastation of modern war is horrific, and no amount of wealth can fully protect you from the worst of it. Even if you stay on top, you'll be constantly looking over your shoulder and always be in danger of someone else taking everything from you.

The rich people who want to live forever will look to minimize these terrible conflicts, which will require minimizing unhappiness, which will likely mean not monopolizing immortality. If most people are older and happy with their lives, they won't threaten the system as much as most people being young, mortal, and miserable. In the short run, the rich will be foolish and terrible with immortality, but not in the long run. The ones who realize the futility in trying to get a free lunch for their own selfish interests will be the ones that live the longest. Selfish assholes will eventually go extinct amongst the immortal rulers, even if it's for a cynical reason.