Genes do have an 'overpopulation' mechanism, it's called ageing.
Well, no.
Ageing has a lot of causes. One primary cause is telomere erosion; this allows a certain number of duplications before the chromosomes get so short that they can't reliably duplicate any more. There are ways to prevent this. But if you prevent this, then you often get cancer. Telomere erosion has a huge benefit - if one cell gets an oncogene activated and starts reproducing out of control, it only gets through a certain number of replications before it hits the erosion limit and dies. This is called the Hayflick Limit. This is relevant here because it both affects our genome and is controlled by our genome.
Now, occasionally a cell gets all the right oncogenes turned on to start uncontrolled replication AND gets a mutation that effectively turns off the Hayflick limit. That cell will now live forever, given good conditions. This, of course, is deadly, and will kill you rapidly from the resulting tumors.
It is worth noting that there is an "immortal cell line" that came from the tumor of a woman named Henrietta Lacks. Not only did these tumor cells not experience telomere erosion (true of most virulent cancers) it also could reproduce well outside the body, as long as they were provided with oxygen and glucose. These cells are now used for all kinds of medical testing, because they are 100% human cells that just divide a lot, and do so forever.
So evolution is constantly in a battle to keep an organism alive and fertile for as long as possible. If evolution comes up with a solution that allows longer cell lives, then cancer becomes more deadly and kills the organism before it can reproduce. If evolution comes up with a solution that shuts down cell division earlier, then you get early death, and the organism can't reproduce as much. There is no "one right answer" to this problem, which is why organism lifespans vary so much.
So no, aging is not a "feature" of our genome that does something useful, like prevents overpopulation. It is evolution's answer to cancer.