I'm not sure your really have a good grasp on what any of these laws actually do, let alone 'free speech'. You keep alluding to complications that don't exist and don't explain what you're talking about.

For "press freedom" defined as the writing and publishing of articles, it's unaffected.

For "press freedom" defined as literally any action a member of the press takes, sure it's affected, but that's a very flawed definition. It's not like drunk driving laws are an infringement on "press freedom", even though they occasionally affect members of the press.

Or perhaps, that USA Today is collecting tracking data on its users and selling it secretly to 3rd parties and doesn't want to stop spying?

Plenty of other newspapers have had no problem implementing GDPR. It's as simple as not spying on your users.