I don't like this idea. Why? Because it would be very ineffective, for a number of reasons.
As is pointed out, you can be 10, 12, 13, whatever, and be really reasonable and mature, or 18, 20, and have whiny little fits like a five year old.
As well, a full number tells nothing. There's something in aging called transition. I mean, 17 years and eleven months isn't really that much different as 18 years, but I'd still be considered 17, now, wouldn't I? A system like you've suggested doesn't take any kind of transition into account, nor is there any really effective way to take such transition into account anyways.
Anyways, we've got the staff around for a bunch of reasons anyways. I mean, they're there moderating chat (as well as modding maps, helping new members, making sure no one's having a fit, etc.). If someone's not acting appropriately, a staff member can address the situation easily enough, and it's not that hard to point it out to a staff member if no one's really hanging in chat much at the time.