I just watched the movie again after a few years. Something that really struck me and also did on first viewing was how quickly the Enterprise crew accept that older Spock and Nero are from the future.
The very idea of time travel being possible they do not really question much. While the movie was seemingly geared to new audiences this is an aspect that felt like it was build on preconceived notions of Star Trek. A big one is that Time Travel is possible.
This alone could be sign of ENT being their past. Even if the Temorpal Cold War was not widley know or believed by the 23rd Century, Time Travel as being possible would have been more seriously considered than it was in Archer's time. Where Vulcan's completely denied it.