The movie is based off of a book written by an American.
The movie was directed by an American.
The movie was produced by an American.
For the movie to exclude this without it being a deliberate decision would be an oversight or coincidence beyond reasonable belief.
Yes, American flags appear in other movies. Captain America is a fictional character in a fictional universe. Use of the American flag in that context is not in the least bit comparable to the first time humans stepped foot on solid ground that wasn't named Earth.
The moon landing was also an iconic benchmark for world civilization as whole. Given that we have concrete knowledge of China forcing studios and actors to parrot China's values, it is certainly plausible that China did not want the crowning moment of this iconic human achievement to prominently feature America at that moment.
It's also plausible that foreign sentiment against the US at that time--not just from China-- was also a factor.
It is also possible the film creators wanted to present the moment as an achievement for all of civilization, without a focus on the US in particular.
But given the history, book, director, and importance for the US, I absolutely believe it was deliberate, and as such people upset about it do not represent a manufactured controversy. If you grew up during the height of the Cold War and the arms race with Russia and watched the moon landing, or have heard--as I have-- stories from my parents and grand parents how much that gave them hope during a very dark time, you also might be a little upset that your country's role had been erased in the crowning moment.
I can certainly understand your point of view and that there might have been an honest choice for that creative direction. But I also think it is completely understandable, and not manufactured, that some people were very upset by that decision, a decision that be could nothing other than deliberate.
I also think it's possible the decision was made for less than purely creative decisions. It's a business, decisions are often made on the basis of money, and it's not unreasonable to entertain the possibility that this particular decision was monetary in nature.
We know Hollywood gets tax cuts for bigger, more numerous and slower-mo shots of American flags in their films.
Clearly "First Man" was making a statement about this special treatment.