and typically should sound much better than direct analog connection.

What would that improve? Builtin DACs already do a great job unless you are going to sample/process the output again in a studio environment.

input the PC audio into a home audio receiver/amp and use the headphone output from that to drive the headphones?

You not only can but depending on your amp it probably outperforms the average headphone amp in every aspect.

Also decent loudspeakers + cheap amp (+ cheap subwoofer) will likely outperform your average PC-Speakers.