Here's the issue with U2F as being the only method of authentication: there is no verification that you are the rightful owner of the U2F device just because you have it in your possession. If U2F was the only form of authentication for everything, that would be like having one master key for your car, house, safe, safety deposit box, office door/building, and everything else. In our metaphorical situation, if someone stole that master key from you, they would have access to almost everything in your life.
Now, you may view my metaphorical "master key" scenario as being the same as a password, but this is the reason why people are encouraged to use several different passwords. Using different passwords is like having several different keys. Rather than having one key that unlocks everything, it's better to have different keys. If only one key is stolen from you, the thief only has access to whatever that stolen key works for. (e.g., if a thief steals your car key, he/she can only access your car.) This is still an unpleasant situation, but it's far better than that thief gaining access to everything you own. Likewise, people are encouraged to use different passwords so if a "thief" obtained one of your passwords, he/she would only have access to a limited number of your accounts.
Using only U2F is essentially comparable to using only one password for every website and using only one key for everything in your life. It's neither reasonable nor secure. As such, U2F should only be used as a second form of authentication in a two-step authentication structure.