Tom Scott polled this years ago, good video. https://youtube.com/watch?v=ALy6e7GbDRQ

Spoiler:

It’s a good thing to question your beliefs every once in a while. Sometimes people get things wrong, and sometimes ideas need re-evaluated.

The way I look at gay/trans/etc is to take MLK Jr’s advice: judge people by the content of their character.

Its a difference in connotation, which is highly subjective and may or may not exist at all.

To me, “Not to mention” implies a bonus thing to consider that doesn’t need to be mentioned to form a complete thought; it’s just additional context.

On the other hand, “Not to leave out” is a crucial detail that needs to be there for everything that was said prior to make sense.

Both sayings are still mostly interchangeable, but connotation is the reason that so many synonyms exist.