> For example, truth is an affirmative defense to defamation in the U.S., even though it's closely connected with free speech rights.

This is wrong, at least on matters of public concern, specifically because it is connected to free speech rights, both for public figures and private figures as plaintiffs, who bear the burden of showing falsity rather than the defendant having the burden of showing truth.

See Philadelphia Newspapers v. Hepps, 475 U.S. 767 (1986); New York Times v. Sullivan, 376 U. S. 254 (1964).