Personally, I think that the reason Michael ultimately decided to have Fredo killed wasn't just because he "took sides against the Family" but because he lied about it, willingly, and then went on his "I was passed over" rant. That rant is what sealed his fate because it showed Michael that Fredo would never be happy just being "the Don's brother"; he would always resent the fact that he wasn't made the Don, whether or not that was "how Pop wanted it". That, IMO, is the real reason Michael had him killed: he could never trust that Fredo wouldn't seize any opportunity or sign of weakness on his part to take another stab at control in the future.