What you quoted "It is important to note that the university did not deny the First Amendment rights or academic freedom of professors..." comes from University of Florida and as far as I can tell is not an airtight argument. No valid conflict of interest is presented in the article for for example.
If typically has allowed such outside work, if there was not a conflict on a governmental issue, then the University of Florida as a governmental institution is performing 'self dealing', in this case by putting their thumb on the scale of public discourse, by limiting/restricting the speech of the professors.
Part of why we have the first amendment is to limit the ability of the government to effect the course of public discourse and the University of Florida is definitely doing that.