It's not so much running 16 bit code, but running something that wants to run on bare metal, i.e. DOS programs that access hardware directly. Maintaining the DOS virtualization box well into the 21st century probably wasn't worth it.

> The 64-bit builds of Windows weren’t available immediately.

There was a year or so between the release of AMD-64 and the first shipping Microsoft OS that supported it.[1] It was rumored that Intel didn't want Microsoft to support AMD-64 until Intel had compatible hardware. Anyone know? Meanwhile, Linux for AMD-64 was shipping, which meant Linux was getting more market share in data centers.[1]