And thus you detected the field of "space situational awareness"...
How do we assure an orbit doesn't interfere with other orbits?
-> We don't.
Dedicated organizations (civilian and/or military - e.g. "18SDS" (US), EUSST (EU), GSSAC (GER), French, Spanish, Chinese, Egyptian and plenty others) do basically in short exact what you suggested: propagate orbit of every known object in the near future and crosscheck every orbit against another.
If a close conjunction is detected, the operator of the satellite is informed and in critical cases risk mitigation operations are performed.
Unfortunately due to orbit perturbations, only looking few days into future makes sense, so these checks are done around the clock.
US Space Force (18SDS) is sharing most of the results in public: see "www.space-track.org"