@weaverjk: I really appreciate you taking the time to get feedback on this topic. Its a grey area. My opinion is more on the side of adhering to the age limits - as a result of bad experiences with underage participants in the past - but the chances that anyone is going to be able to tell that your 12 year-old is not 13 is very slim. If they can "pass" - and especially if, in addition to games with family and close family friends, they have experience playing RPGs with other adults and older teens - no one is likely to have issues at the table.
In my opinion, when it comes to RPGs, the age limits are less, I think, about chronological age and more about maturity. With boardgames, age recommendations often attempt to communicate complexity whereas with RPGs, I think age limits on convention events are more like MPAA ratings that attempt to communicate maturity and theme. YMMV.
I'm in the camp that would prefer *not* to play with obviously younger children. A 12 year-old in a 13+ game? I doubt I could tell the difference - especially if they carry themselves with more maturity. Unfortuntately, I've been in games where parents have brought *much* younger children to the table with a "They've been playing for years. They know the system." and those games did not go well.
While younger kids may have known the system mechanics (possibly), what they didn't know was how to play well as part of a group or to play well with others significantly older than themselves. When the styles of play at the table are not aligned and the players themselves do not have the maturity to recognize and align them as needed, the table experience goes sideways very quickly. Mature players are able to make those adjustments; immature players not so much. (As others note above, age and maturity are not always aligned - I, too, have seen chronologically older players who behaved like young children; I've played games with astute children as well...but, outliers aside, we should all be able to agree that there is a correlation between age and maturity).
For parents who are considering bringing underage children to an event, please consider not only your experience and your child's experience, but also consider the experience of the other participants at the table. If you are coming from afar, Gen Con is not a cheap trip - either in terms of time or money - and a paying participant certainly has the right to expect to have fun while there. If your child has no experience playing with previously unknown adults - like at free games at your Friendly Local Game Store, Pathfinder Society, Adventurer's League, etc. - maybe breaking them in at a paid convention like Gen Con isn't the right thing to do?
Its a tough one. I know that I certainly want to encourage the next generation of gamers - but you have to consider the fun of the other players at the table as well.
While 13+ might be a gray area, my opinion is that 18+ and 21+ events are not for kids. Either because the themes of the game are in the "R" area or because the GM is, as clear as possible, communicating "No Kids". Like others, I *do* look for those events precisely because I know they are likely to have few if any children at the table.
Finally: GMs - its your table. For 13+ events, I'm not opposed to younger players at the table within reason (see above) - but then its your responsibility to ensure that the spotlight moves appropriately; obvious weirdness, off-premise, off-genre play is tamped down; and that ample opportunity is given to other players to interrupt inappropriate behavior from the less mature players (regardless of age). Allowing an immature player to cause chaos at the table ("I spit in the King's face and tell him he's ugly", "I run up and grab the loot", "I steal the other player's magic ring", whatever) without redirecting (or rejecting) their behavior is not cool. (Sorry, gang, "Its what my character would do!" may work in your regular gaming group at home, but at a con with a bunch of strangers you need to find ways to work together - especially since most of the time, the characters are pre-made and supposed to know how to work together anyway). Remember: we only have 4 hours to get a group of players with potentially different play styles, expectations, and preferences to come together as a group, find and bite the adventure hook as a group, go on and complete a mission as a group.