Actually, Care.com IS confusing when it comes for the nanny to giving consent about the background check. Basically there are 3 ways for the nanny to get there, and 2 out of 3 will require nanny to pay $ that the family already paid for (but then the nanny owns that check).
OP, please ask your nanny to to go the message in her mailbox on the site and follow links from there. If she tried to do it from her profile it will require her to pay. It took my nanny a few hours to figure it out, and I'd consider her technologically OK. It is just Care trying to make extra $ on the nannies.
For previous nannies I ordered a more in depth investigation than Care does and did not see too much differences -- and honestly all of them are limited to "whether the nanny was caught with any crimes" or not. All the good checks (including Care Preferred checks) include a check by county where the nanny lived whether there was any record with local courts. It is a necessary thing to do -- and I agree not sufficient -- but then it is not a more in depth check but more references that I rely on. If your potential nanny committed a crime (e.g., stole something from a family she worked for), but was not caught/accused in court by the family there is no way you'll find out about it from any check.
The nanny that is less tech savvy could be a plus in my -- less facebooking and texting while she watched your child
