Yes, it would, but Bilbo makes an exception to the rule. The language around events is more explicit than the language around conditions.
The language around conditions is written as an affirmative, leaving open other times that they could be played (if another card allows it). If the rule said, "Free Peoples conditions cannot be played in any phase except the fellowship phase and Shadew conditions cannot be be played in any phase except the shadow phase," then it would be an exclusive negative statement.
The language around events is written as a negative, excluding all times except the appropriate phase.