Tried some VxD drivers on Win98 and it doesn't help, either. Same behavior -- no sound output, FM/MIDI produces nothing but pops.
Whenever Device Manager refreshes (manually, or when making changes to devices), pops can be heard from the sound card's output.
Additionally, when I try to remove or alter (update) the main device ("ES1869 Plug and Play AudioDrive"), along with an audible pop, Win98 would raise a 0E exception that can be recovered.
On the other hand, everything works fine with this card on NT 3.51 using this ES1868 driver which can be found here. Both ESFM and MIDI works as well as playing 8-bit and 16-bit audio.
EDIT: I've found the answer -- for this card, even when using a ES1869 chip I still have to use the ES1868 driver. ES1869 driver won't work for this card's setup.
For now ES1868 VxD driver works fine with this card. I probably have broken the system's WDM functionality so I cannot verify whether ES1868 WDM driver works (the driver installs but multimedia properties report nothing present).