"Once we've had our children and raised them to be independent, evolution doesn't really care what happens to us and there is little selective pressure to prevent cancer-causing mutations."

That is true for some species, but it less true for social animals where grandparents play an important role in delivering resources to grandchildren. Your statement is least true for humans, where grandparents play a very large role in delivering resources to grandchildren.

For our size and weight, humans are unusually long-lived, and probably a reason for that is that grandparents do play a role in the survival of grand children and even great grand-children. Thus, among humans, evolutionary forces are still at work even late in life.

Also, keep in mind immortal humans may want to bear children at ages much later than current ones can. For instance, it would be helpful if you could bear children at 30 year intervals so you could be sure you raised your kids right (or try a different approach if you didn't)

As to having children later in life, that could certainly help the situation, but only if we all agree to do that. If there is some genetic factor that determines when a person wants to have kids, then people who want to have kids earlier will likely win out.