Yes. Apache already has mod_spdy and nginx has plans to implement it. Browsers fallback to regular http if the server doesn't speak spdy. There are no disadvantages for you.
I think programmers have achieved harder tasks than detecting which protocol the server speaks and speaking its language, starting with the most optimal protocol and falling back to simpler and more common ones.
Is Mozilla acting in the interest of the Open Web or was this a bargaining chip with their $300M default search provider deal?