I've used Twilio numbers for this purpose in the recent past. Those work 100% of the time so far (assuming the number hasn't already been used).
Many services seem to run checking on whether a number is really a mobile number or not. VK (the Russian social network) for one example is a zealot about that and won't let any Twilio numbers through. And Instagram is a lot tighter (better at fake account creation detection) on authentication than Twitter but you can still sign up for Instagram with just an email account in most cases.
To get the individual user information for a Twilio number, given the short duration they might be attached to a given account, I highly suspect you have to go through Twilio for that info. That is, Twilio is likely to be the sole source for information on which account is attached to what number at a point in time. Obviously any consequential government authority in the US can then get that from Twilio, assuming the required judicial merchandise is presented.
I'd be curious though to know if eg Twitter security sends a request over to Twilio (perhaps in a case of an account abuse investigation), if they'd cough up information very willingly. That I don't know. It's a certainty the government can get at it, however. The best bet is to do nothing illegal using a Twilio number, and to assume if a big external service provider (Twitter, Facebook, et al.) needs to get details for some legal reason, they will.
Many services seem to run checking on whether a number is really a mobile number or not. VK (the Russian social network) for one example is a zealot about that and won't let any Twilio numbers through. And Instagram is a lot tighter (better at fake account creation detection) on authentication than Twitter but you can still sign up for Instagram with just an email account in most cases.