Saving the current state? What does that even mean? You don't save transient things like visual particle positions but it's something you expect to persist between pauses.

Pausing the render? But not the physics so you keep falling? You need to pause many systems. At the very least you'd want to pause gameplay, and sound and physics. You'd want to keep input and some of the UI unpaused. If you have a fancy UI with VFX, you need to make sure those are not using the paused game time. etc etc

It’s only simple if your state machine has zero implicit state and all transitions are perfectly and precisely articulated. Good luck with that!

P.s. And once you are done achieving the above, you can then make sure you haven’t caused performance issues :)

But yes, conceptually, it’s a relatively simple idea. The devil is always in the details.