If he had not been a film star, he would not have become President.
In my opinion, "a President" is correct. Examples:
"My dream is to be a teacher"
"My dream is to be a President"
Which is correct usage? President or a President?
If you are referring to the President of the United States (or of any country, for that matter), you have to bear in mind that there is only one of them at a time. Therefore 'President', without the indefinite article, is sufficient to describe him (or one day, her): it is an official title as well as a descriptive term.
If someone says "I want to become a president" or "My dream is to be a president", the implication is that they don't care which country they become president of.
The article "a" indicates the noun is one of multiple possible exemplars. There's only one President of the United States (POTUS) at a time, so the correct article, if an article is used, is "the".
If I saw "a president", I'd assume the person was talking about a business, since many of those use that title, and almost all have the similar title of "Vice-President".
As for leaving the "the" off... any supposed rules for when "the" is used are simply justifications. The real answer is that when talking about the POTUS in this particular context (to become POTUS) people almost always do not use any article whatsoever, so that is the correct way to say it. English is a language used by humans, not a programming language with consistent rules so that machines can easily verify it.
This sentence speaks of becoming the President of the United States of America:
"If he were to have not been a film star, he would not have become President."
This sentence speaks of becoming any president in the world:
"If he were to have not been a film star, he would not have become a president."