Console-based microtransactions...
For what it's worth, you can delete your credit card details from XBLA, but you can't do it from the console. Instead, you have to log onto the xbox.com website (which uses the same Passport-based authentication details) and delete it from there. It's not a particularly user-friendly system and IIRC, Microsoft don't actually provide any information on how to do this on the console itself.
(in brief: log into xbox.com, click on "My Xbox", click on "My Account", click on "Edit Payment Options". Not so simples!)
Also, it's worth noting that this is a general "kids with parent's credit card" issue - a relative had a similar issue with their son, who ran up a bill into the thousands on his mobile phone, back when there were a lot more of the dodgy "implicit subscription" Crazy Frog/ringtone con artists around. Any system where you don't need to explicitly provide payment details is subject to potential abuse/misunderstandings.
However, I'm not sure Microsoft can be let off so easily. They already have a set of child-protection/restriction systems in place: by default, kids are barred from video comms, their friends list is hidden and they can't view "explicit" content. Surely it's a no-brainer to add an extra "billable transactions" option and provide a set of options:
1) no billable transactions allowed
2) billable transactions with stored credit card allowed
3) billable transactions allowed, only with XBL points card or manually entered CC details
If a default choice of 1) is set, that would seem to cover all of the bases: if a parent decides to let a kid have 2), then they can't complain if a bill gets run up. Meanwhile, if they select 3), then the kid can buy a points card from HMV/Walmart; if they go a step further and steal mummy's credit card, then I'd say you've got a bigger problem than a bill for some pointless avatar items...