While it might have been possible to add PC speaker music, would it have been worth the effort at the time? How much would it have delayed the release of the game? Would anyone have cared very much about being able to have terrible quality music on their PC speaker?
So they'd have had to extract this feature from a vendor where the relationship is already breaking down, or ditch it all and start from scratch with a new vendor, or inhouse code.
This is just someone cosplaying early 1990s development with the benefit of decades of subsequent OS and software design.
[id Software was Lazy - DOOM could have had PC Speaker Music!]
On Linux, `sndserver` is a separate process. On MS-DOS, it was not. On DOS, priority mixing would have to happen e.g. within Doom's tick processing or in the sound interrupt handler.
If you're just making these changes in Linux and assuming they could have worked in 1993, you're just cosplaying.