I feel that so many people that identify as non-binary only do so because they don't conform to their gender stereotypes, not because they have genuine gender dysmorphia.

Actually, the thing about being non-binary is that non-binary can and often do experience gender dysphoria* (which is what it is; not dysmorphia). It seems like an overwhelming aspect of this post is that you base your views off of the belief that being a non-binary trans person automatically equates to being a non-dysphoric trans person and vice versa, but this isn't true.
There are binary trans people who do identify with being non-dysphoric, and there are non-binary trans people who identify with being dysphoric. I am one of the latter.
My gender identity and how I arose did not come out of a desire to be a "man that plays with dolls and likes painting his nails". It came because I ultimately realized that my primary and secondary sex characteristics caused be significant distress, something easy and clear to label as gender dysphoria, but I am absolutely 100% non-binary.

I think that if gender stereotypes weren't so strongly present - if boys didn't feel that they had to dress a certain way, if girls didn't feel they had to like certain things, and so on - then non-binary genders would barely exist.

Once again, gender stereotypes are not what make me dysphoric. I can say it clearly: even if gender stereotypes were abolished, I'd still hate my primary and secondary sex characteristics.

I'm not referring to trans gender people, who I've read often feel like they are trapped in the wrong body, often from a young age.

Non-binary people are transgender. Also, just a quick sidenote that "trapped in the wrong body" is not particularly the most nuanced way to describe the feeling of dysphoria.

I'm referring to people who are gender non-binary, and in particular people who may change their gender on an ongoing basis. Surely, if you were to 'wake up' and feel like a different gender, then it's really not your gender that's changing at all, but merely your mood.

Not all or even most non-binary people are genderfluid, and even to the ones that are genderfluid, it still isn't necessarily as simple as "my mood changed so my gender changed", as far as I'm concerned. However, as a non-fluid non-binary person, I can't speak too much on the experience.

I've had many days where I wanted to go camping and light fires and throw big rocks into the river. I've had many days where I wanted to bake cakes, drink wine and gossip with friends. My gender in both of these occasions was the same - and the fact that me, you, and everyone reading this knows which example above corresponded to which gender is surely the problem here, not actual gender dysmorphia.

This entire last paragraph once again shows that your argument is on a false premise. Like, your entire point is that "people identify as non-binary because of gender stereotypes; not dysphoria!", but I'm a non-binary person who is non-binary and strictly identifies that way due to dysphoria without any regards to any social norms, and get this: I'm far from the only one.