I don't think "He played a major role in ... championship..." means he was an actor in the theater. Maybe the theater of life. And acting a professional role is not faking it. So the real and metaphorical use are similar.

I don't understand your confusion. The definition you've quoted for play a part does not suggest anything about level of significance or irreplaceability. In a play, there are major and minor roles. In a government, there are major and minor roles. In a scientific discovery, there are major and minor roles. On a sports team, there are major and minor roles. I can't see any difference in meaning, or in usage.

en.m.wikipedia.org/wiki/All_the_world's_a_stage considers the long history.

It's (ahem) a Metaphor. "All the world's a stage, And all the men and women merely players; They have their exits and their entrances, And one man in his time plays many parts."

@Yosef Baskin I added the addendum. I would like to know your opinion.

@John Lawler But this expression is used not only in relation to people. Sometimes it refers to inanimate things at all. For instance: "...the role played by diet in disease" (example from collinsdictionary.com).

This is how "play a role" appeared with an absurd theatrical figurative meaning, in which there is no logical connection with its literal meaning. If that's absurd, and apparently without logic, (which it isn't) what would you say about 90% of English idioms and phrasal verbs?! For example, "a different kettle of fish", "under the weather", "pot calling the kettle black", with "put up with something", "take something on" and "catch up on something".

@Xanne But this expression is used not only in relation to people. Sometimes it refers to inanimate things at all. For instance: "...the role played by diet in disease" (example from collinsdictionary.com).

This is language; speakers compare a familiar situation "play a game/sport" with "play an instrument" with "play a role" with "playing with words" with "playing around" = having several sexual partners, etc.

@Juhasz There is an effect and influence in the definition. I talked about the significance, because significant things have an effect and influence, and not significant ones do not.

@Mari-Lou A If this makes sense for you, please explain it. My goal is just to understand. I explained in my question why I do not see the logic. I am not suggesting that English idioms have no logic, I believe they are based on logic. And my goal now is to understand the logic in this case.

Acting is like a game, it's fun, it's also competitive (who's a good actor? Who has the better memory? Who can make the audience cry and believe in you?) An actor takes on a role, they pretend to be someone they're not. Take on a role = play a role. A theatrical piece of work is also called a play. Which came first, playing a character/role or appearing in a play?

@Mari-Lou A I understand that this is a language, not a crocodile. That it is a language is not an argument. A metaphor is based on a presence of common properties, and not on a fact that things are familiar to someone. I want to understand what are common properties in this case. In my opinion, some properties contradict the figurative meaning.

I explained what speakers did in the past and today, they compare two, three situations and assign a common verb. Play = fun. Play–game, sport, music, with words, a role ("a part" on stage and then a role in life), around etc.

@Mari-Lou A I have read your comments. Thanks for clarifying your understanding. I'll think about it.

When actors accept a job playing a character, they are not playing a part, they got a job. Best practice is to take words at their plainest meaning, not in an expression, not as sarcastic, not as an abstract. So if I act in a show, I may be playing a part in the grand scheme of things in the arts, but first I am playing in that show because I got the part. I start with 2 and 2 = 4, not 22.

@Mari-LouA The definitive answer to this question, which has not yet been posted here in comments or answers, can be found easily enough by anyone with access to the extremely detailed but paywalled OED entry for the verb play. Those without direct access may be able to gain OED access through their nearest city library or university library, at least in English-speaking countries. You usually can in the UK, and I know you can here in Colorado via the University of Colorado's library from when I was an RA.

As I said, it's a metaphor. Metaphors are not logical (my love is not a red, red rose, in fact) and deny reality. Nothing new here. But they're the only thing we have, so we use them. They make sense, but not logical sense.

@tchrist There is Oxford Dictionary on lexico.com. What are the advantages of oed.com.

@John Lawler The logic of a metaphor is in the presence of common features. I think in "A red, red rose", they might be as follows: 1. A red rose is beautiful, and a beloved girl is beautiful, and love is beautiful. 2. The red rose blooms in June, the author's love arose in June. 3. A rose blooms once a year, and true love happens only once in a lifetime. 4. A rose is beautiful, but fragile and fades over time, just like the feeling of love can be beautiful, but it is fragile and can fade. 5. Pink is associated with love, "red, red rose" (concentrated pink) is strong love.