It'd make the chips easier to design, I guess. Maybe motherboards and BIOSes too?

I could imagine a more user-noticeable effect in VM cold-start times, perhaps, though I'd think physical hardware boot times would still be dominated by other things (e.g. actually loading the kernel into memory); the minimal implementation of transitioning from real mode to long mode is a few hundred instructions.

Save die space and thus reduce cost and power budgets. Potentially simplify microarch and cache management. Maybe reduce attack surface?

Nah, the die space you save here is absolutely minuscule. Intel saves on the validation of core designs -- forever.

It might not make too much of a difference to software developers targeting x86, but dumping legacy crap might make Intel's job easier (and/or more profitable, because they can reclaim some die space) when it comes to implementing their future silicon.

A cynic might wonder if it would also help to renew the Intel/AMD x86 patent moat now that the core x86-64 specification is over 20 years old. If the plan is largely just to strip things out it might be hard to shoehorn in patent claims, but I'm sure a way could be found.