Executing 16-bit code in Compatibility Mode (not Long Mode) is possible, that's not the problem. The problem is lack of V86 allowing legacy code to run. So Real Mode code is out wholesale (a sizable chunk of legacy software) and segmented memory is out in Protected Mode (nearly the totality of remaining 16-bit code).

So yes, you can write/run 16-bit code in 64-bit Compatibility Mode. You can't execute existing 16-bit software in 64-bit Compatibility Mode. The former is a neat trick, the latter is what people actually expect "16-bit compatibility" to mean.

> segmented memory is out in Protected Mode (nearly the totality of remaining 16-bit code).

No, segmented memory is exactly what you can get working. You set up the segments via the LDT, which is still supported even in 64-bit mode; this is how Wine is able to execute Win16 code on 64-bit Linux. (Reading Wine code is how I figured out how to execute 16-bit code from 64-bit code in the first place!)

What doesn't work, if my memory serves me correctly, is all the call gate and task gate stuff. Which is effectively building blocks for an OS kernel that everyone tossed out in the early 90s and instead went with kernel-mode and user-mode with the syscalls (first software interrupts and then the actual syscall instruction in x86-64). You don't need any of that stuff to run most 16-bit code, you just need to emulate the standard Windows DLLs like kernel, ntdll, and user.