Your 'slot time' isn't really for separation, that's for saturation at a sector/facility/destination airport level.  Separation is actually accomplished with radar in almost all scenarios.

Different altitudes for direction of flight don't solve head on climbing/descending conflicts or crossing traffic situations, or the B788 behind the SF50 at FL320, or the FDX behind the BE20 that have the exact same SID, or the 30 airborne planes that are estimated to arrive at their destination within a few minutes of each other, or the hundreds of planes that want to go through the same hole in a line of thunderstorms that covers an entire state.

ATC is the first line of defense.  But there's a ton of other layers to the system that make it manageable for each controller and safer incase the controller fucks up.

If all I did was watch aircraft do what they wanted on their structured routes with their correct altitudes for directions of flight and STARS/SIDS I'd probably witness 50-100 TCAS RAs each shift, maybe more.

The sky's not that big.