Will using such functions instead of arguments mean lower register usage overall, or how do those things interact with each other across function calls?

It emits semantically the same spirv code as previously: define a builtin global variable and read it. Just that you can now do this anywhere in your codebase, rather than having to pass it down from the entry point and have to hope noone accidentally passes the wrong thing.
Register usage heavily depends on the target platform. See AMD RDNA3 ISA pdf on "3.5. Initial Wave State", there you'll see compute shaders use 4 scalar registers (SGPR, once per workgroup) for workgroup_id plus some bitflags. For local_invocation_id they seem to have been changes in RDNA3, it seems like it only uses 1 vector register (VGPR, once per invocation / thread) now (even though the table in the pdf defines it for GS (eg. mesh shaders) not CS, I assume it's the same), whereas RDNA2 used 3 VGPRs, one for each axis (see "3.12.2. SGPR Initialization").