WINE on Linux x86-64 will run 16-bit Windows programs, by the way.
However, you cannot do virtual 8086 mode without switching back to 32-bit mode, so you cannot run real mode MS-DOS programs. This would probably have broken a lot of 16-bit Windows software that still relied on real mode components, so it does make sense for this and other reasons that they dropped support for 16-bit software.
You can switch to 32-bit mode, then enter vm86, and then switch back to long mode once you are done. There was a kernel module flying around that would do this and make syscalls such as vm86plus work in amd64 Linux.
But yes, basically, it's a political more than technical choice to drop support for 16-bit programs on 64-bit Windows; same for >4GiB RAM support in 32-bit Windows.
Actually, now that I think of it, having KVM capable of running real mode DOS is a rather recent development. Looking at you, AMD, but even early Intel VT-x didn't get it right until 2010ish (Westmere).
Anyway, dosemu2 does exactly that (use KVM to run a lightweight kernel that just enters vm86).