You can enter a rating into your profile and it will be displayed.
However, without some form of verification, anyone can claim to be someone else, in your suggestion. Short of verifying titled players and for members of the US Chess club, that is unlikely.
I'd agree with you if chess.com didn't already have all the information it needed to ensure that didn't happen. Once submitted, chess.com would make sure name, dob... on file with chess.com MATCHED name, dob... on file with USCF/FIDE. That's not completely fool proof, but close.
The site doesn't verify DOB or anything other than the e-mail account on anyone but titled players. The USCF also doesn't provide DOB on the membership records, at least not accessible to normal people. FIDE does but I can go look up any FIDE rated player and get their age and claim to be them. I can also put in their name as my name.
It is simply not possible to do without member vetting and that takes resources that the site could better focus elsewhere, in my opinion. As it stands now, members can self record their ratings. I have mine up, though I haven't updated it in a while 
edit: the default site doesn't have a DOB field or request it at any point in the registration process as far as I'm aware.
I think it would be great to have USCF and/or FIDE displayed with all our other ratings (blitz, bullet, daily...). Do you think that would ever be possible? It'd be a really nice, convenient way to see how our online rating(s) relate to our true OTB ratings. I was just thinking your engineers could figure out a way to add this feature. To make things a little easier for chess.com engineers, perhaps chess.com could simply create a form that we, the individual users, would fill out (including our USCF/FIDE number), submit, and then your system would be instructed to display current USCF/FIDE rating (and also instructed to update, say, on the first of each month).