isn't ctrl+a the BASH default for 'go to the beginning of the line'?
It is. If you're using screen, you can press ctrl-a a to send ctrl-a to readline to go to the beginning of the line. ctrl-b will actually move back a single character. Presumably, you could do something like ctrl-b b in tmux to move back one character, but I haven't tried it, so I don't know if that would work.
They didn't want to clash hotkeys with screen when tmux was first released.