Since bitcoin transactions, no one trace and bitcoin gambling sites don't want to do any verification about the players. Because that will affect that bitcoin's anonymity function. So I don't think these bitcoin casinos can put any restrictions on any underage people not to access the site. It is players responsibility not to go for gambling if they can't control themselves on when to stop.
If we assumed that online gambling sites required verification than mostly players will be avoid to play online gambling, also me i avoid to give the personal information in online gambling sites. I don't think that it should be necessary, without verification we can play online gambling than it mean it is not a necessary. But your point is right, it is our responsibility not play gambling in a bad way.
I will also not support for this Bitcoin verification, to avoiding this we are using bitcoins to play gambling. If they introduced this verification method, then no people will use bitcoin to play gambling. And it is not possible the casino will not ready to lose their business, so they will not introduce this method. And i agree that we must play gambling in right way, don't choose this game for make money.