I think freedom and universality must have some deep connection.
If you have the freedom to do something, that means -- wrt that context -- you can do whatever you like. You're unconstrained; that aspect cannot (or at least should not) be a bottleneck.
That gives you some unboundedness wrt method, or content, or whatever the freedom relates to. Freedom of speech removes a bound on speech. Freedom of thought removes a bound on thought.
Freedom enables universality; and without certain freedoms, you cannot have certain universalities.
I thought that freedom and thinking were like orthogonal; both necessary for philosophy and good life, but neither sufficient. Now I suspect there's more there than I realized.