As someone who doesn't have kids, and as someone who finds how a lot of companies bend over backwards for those with kids quite unfair, I can see how this dishonesty manifests itself.
Though, this is technically not your issue to deal with.
I'm glad you are being honest about it pissing you off. It would piss many people off, and I don't think people realise that this also needs to be resolved. It may be quite difficult to watch people you respect being lied to.
You need to distinguish fraud, where the employee gets a company sanctioned benefit, where you may have an ethical obligation to inform your employer, and just peer-dishonesty, which is not actionable by your employer.
I think your first step should be to even the playing field, and ask for the same benefits that he is getting. If everyone starts asking for extra time off, management will have to publish a policy regarding who can claim these benefits and what is required for proof.
Don't guess what he really gets up to. The comment about computer games is irrelivant.
If you do feel inclined to raise this with management, I would do so anonymously. And yes, I personally would consider letting your coworkers know anonymously if nothing is done and the behaviour doesn't change. Though I fully understand if me saying such things will get this answer a torrent of down-votes.
Whatever you decide to do, your focus needsmotivations need to be on stopping the dishonesty, and not punishingtrying to get the employee punished for it, as much as you want to.