if in your mind he's doing a good job cloaking hate in free speech rhetoric, how can you be so sure that he isn't actually arguing the free speech case? Hate isn't a subtle emotion, and Dr Peterson's arguments are reasonable to those who aren't fully committed to the postmodern understanding of sex and gender. Both sides of the debate are perfectly reasonable to hold and the issue isn't in any way settled (at least in part due to how politicised the study of gender and sex has become) so to suggest that Dr Peterson hates trans people because he disagrees with the far-left narrative on gender is misguided at best. If you think his views are outdated then that's a fair criticism but it's not like the current postmodern perspective on the issue is old and well-established.

> how can you be so sure that he isn't actually arguing the free speech case?

A fair point. But I'd suggest that if he was just actually arguing the free speech case, there would be no need for arguing whether trans people are a legitimate thing (cf his comments on biological sex <=> gender causality). His broadening of his argument to that and "leftists are trying to force new words into the language" (cf Canadian Senate testimony) suggests he's not actually entirely about the free speech but rather just finding a fanatical new audience for his railing against SJWs and snowflakes and political correctness.

> to suggest that Dr Peterson hates trans people

I'd say "anti-trans" rather than "hates trans" - there can be a world between those two viewpoints.

it doesn't seem to me that he's saying trans people aren't a legitimate thing. My read is that he's saying that gender is related to sex in the sense that your gender is somewhere in a spectrum of male-female (in a similar sense to masculinity-femininity). Which every single trans person I've ever met or spoken to fits into.