Maybe people on the right are less likely to seek out help from mental health professionals. People on the left have less reservations about it, and therefore see more diagnoses.

I'd bet they assume anything they don't like is mental illness. Probably anyone LGBTQ is mentally ill by their definition. Anything they agree with is mental prowess though. People can hold whatever opinions they want, but some are more valid than others, and this one is not valid.

if you try "leftists (or liberals) more likely to be mentally ill" in a search engine you see things like:

American adults who identify as politically liberal have long reported lower levels of happiness and psychological well-being than conservatives, a trend that mental-health experts suspect is at least partly explained by liberals’ tendency to spend more time worrying about stress-inducing topics like racial injustice, income inequality, gun violence, and climate change.

https://magazine.columbia.edu/article/why-depression-rates-are-higher-among-liberals

Actually, he's right. A quick Startpage search offers this.