Unfortunately, this "illusion of time pressure" in the main story, while you are supposed to do side content first, is a problem many games have. This creates a disconnect to me. Not sure what a good solution might be - perhaps building in some major breaks into the main story, so you don't feel bad for doing side content?
You've got the solution right there. You write in clear, explicit breaks to allow for side content without a weird dissonance. Morrowind outright tells you at points in the main quest to go and get more experience in the world while your handler dude awaits intel/communications with superiors. It's an elegant solution.