Things like Linux should work fine with a CPU that strips 16 bit mode entirely (32 bit too, possibly? not sure) as long as you have a BIOS / boot loader that can handle it and - as of when I last looked at the Linux kernel initialisation code over a decade ago - change / strip out a handful of lines that took care of changing the mode.

It'd be interesting, but I don't think it'd save all that much unless you strip 32 bit compatibility as well, and even then it might be less than you think or they probably would have tried to see if the market would want it...