Only if you build tunnels that no one can get to. Road tunnels still need to obey grade and (horizontal and vertical) curvature restrictions. For example, every story down you push a tunnel, every city block longer an access ramp needs to be. And the deeper you make tunnels, the more difficult ventilation and pumping becomes.

theoretically one could avoid vertical curvature issues by connecting the tunnels with large shafts, similar to elevators in skyscrapers.

This may slow down traffic, based on the vertical speed of the elevators and how quickly people in cars can board the elevator.

I'm not sure which one would be quicker: having a really long access ramp (long because of the tunnel's depth), or having to wait in line to board an elevator that moves quite slowly.

Musk's system uses elevators.

Now figure out how to get people up and down, and how to cool them. Places like London has problems with both already.

Look at Paris, this is not true.