What is the origin of the phrase "play a part/role"?

From Longman Dictionary of Contemporary English:

play a part/role
to have an effect or influence on something

Does this phrase come from the theater or somewhere else?

From thefreedictionary.com:

play a role in (something)
2. Literally, to have a job portraying a certain character in a performance.

If a person is an actor and plays a role in a theatrical performance, this does not make him an important person. In a play, he can be replaced by someone else, maybe even by someone better than him. In addition, there are simple, insignificant roles that anyone can play at all, not even a professional actor. Thus, playing a role in a play does not mean being important, irreplaceable, and influential.

Moreover, "to play a role" means to be not real, because a person does not play himself on stage. He plays a literary character. He is not himself.

However, this phrase is used in a serious way to mean involvement and influence.

Is this a metaphor? What is the comparison here?

From theanalyst.com as an example:

He played a major role in Bayern’s ninth consecutive championship
during the 2020-21 Bundesliga, as the only player with double figures
of goals (11) and assists (18) in the competition. No other player was
involved in as many open-play sequences ending with a shot as Müller
(186), with 14 of these being the final pass before a goal.

Literally "He played a major role in Bayern’s ninth consecutive championship…" means: He was an actor who played a major role in the play "Bayern’s ninth consecutive championship". He is not himself because he played a role. He’s just an actor, nothing really depends on him. The play does not change the plot if the major role is played by someone else.

It turns out that the figurative meaning of the metaphor is almost completely opposite to the literal one.

However, a metaphor must use an analogy, a metaphor is not irony to use an opposite meaning.

What is the etymology or origin of the phrase «play a part/role»?

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Addendum

According to etymonline.com:

Meaning "to take part in" a martial or athletic game is from c. 1200.
It has been opposed to work (v.) since late 14c.

Thus, before its theatrical meaning, the word "play" meant participation in a competitive game.

Could it be that "play a part" originally referred to competitive games? "Play a part" could mean to participate in a competition individually or as part of a team. In this case, the person is himself, and victory or defeat really depends on him.

Then, with the growing popularity of theater, "play a part" was mistakenly associated with a performance of a role. "Play a part" in a competitive game was confused with "play a part" in a theatrical performance. This is how "play a role" appeared with an absurd theatrical figurative meaning, in which there is no logical connection with its literal meaning.

However, this is just my hypothesis.