Trans* advocates do not advocate any new type of restriction on speech. They do sometimes argue that restrictions that already exist on speech when that speech targets specific types of groups should also apply when such speech targets trans* folks. This is not in any way stretching what limitation on the rights of free expression the constitutions of Canada or the US (or any other country) will tolerate. That doesn’t mean that there aren’t serious threats to free expression in the US and Canada today.
I’ve written previously about how changing “loser pays” presumptions in defamation lawsuits can impact the related freedoms of the press and personal expression (though focussing primarily on the impacts on press freedoms). But compromises around such details as when the plaintiff has to pay a press outlet’s legal costs aren’t the only constitutionally permissible actions that the governments of the US or Canada might take that would have an impact on free speech, and there are even other, likely impermissible actions that the government might still try to take.