Letting nature control population means relying on one or more of nature's methods. These are also known as War, Famine, Pestilence and Death. Human choices of control are preferrable if they beat nature's. By beat, we could be talking about "less wasteful", "kinder", or somehow "ethically fairer", and the exact conclusion will vary depending on which we emphasize. In fact, we could be trying to balance many such goals. You may be arguing from some definition of "ethically fairer", "less wasteful", "less arbitrary", or some other standard. So if you really want somebody to tell you what's faulty about your proposal, until you can explain what you are trying to accomplish better than by just letting nature take its course you don't really have any logic behind your claim to refute. Without that understanding, your proposal is an emotional argument disguised as a reasoned conclusion.
Knocking out aging actually has relatively little effect on population growth in some ways, for example women still stop having fertile eggs at menopause even if they typically live much longer. How many of those opportunities to fertilize an egg actually get used has a direct effect on population that is really larger than any possible additional lifespan.
(Yes, try the math. Increase the lifespan to a blisteringly worst case full 800 years, which would be about the average if we assume nobody dies of anything except violent accidents and deliberates such as being struck by a bus or shot in a war, and add some additional worst case for population assumptions such as that most of the people who kill themselves either do it early or wouldn't do it at all if they had their health. Assume ALL fatal diseases are cureable, and all people enjoy a biological age of about 25 for as long as they live, but women still stop being fertile about 45 to 50. Now instead assume current longevity prevails, but take the worldwide reproductive rate back up to about 4.2 children per generation, add that we can somehow feed all those kids for a few generations and so the rate can (temporarily, from a long enough perspective) stay that high, and now guess which group eventually gets bigger than the other way.).
By the way, surgical sterilization is seldom reversable. The usual effect is that closing off the tubes (for either gender) triggers internal scarring and often within a couple of years an autoimmune reaction sets in which causes the eggs to become infertile or the sperm to not fully form. The odds of a pregnancy resulting from a successful reversal are as low as 20% for the most common methods of female sterilization, although there is a procedure involving simply banding the tubes with clips or rings and doing no cutting and this gives odds as high as 70%. Male sterilization reversal has slightly better odds than that, but this assumes the surgeon did the original procedure with an eye towards eventual reversal, the reversal can include more than a simple reconnection but be followed as necessary with a complete epididymal repair (with a doctor who can determine on the fly which of three different procedures should be used after he or she actually gets in there) and the auto immune reation didn't happen. We're talking about a great success rate if you have one of a few dozen extremely skilled doctors who can do that work, but those guys are a bit like heart transplant surgeons - they don't grow on trees, and they don't come cheap. If you pay a doctor public clinic wages to bulk sterilize poor people, he or she won't be a doctor with that sort of success rate on reversals. You're making something sound simple and reliable which is actually pretty much experimental rocket science, and nobody should get sterilized with the idea that it can reliably be fixed if they change their mind or circumstances..