I am another with this problem. I have switched my Xbox off and on again, cleared the cache more times than I care to remember, deleted and redownloaded my profile (whilst clearing the cache what felt like another 100 times). I read the troubleshooting page which said my account may not be secure enough. This could be the problem as my Xbox has been telling me this for months. However, I have given every email address and phone number I have to "make my account more secure" and I have quite a few. I have even done this multiple times, but I just tells me I have already entered that particular address/phone number. I have no idea what else I can do.
My profile won't even register offline, meaning any game I try to play cannot read an active profile. This means my 360 is now pretty much useless.
My Xbox one does not have any problems signing into the same account, and no security issue has ever been mentioned on it.
Any help would be appreciated