The reason audio vanishes is because the routing gets a little buggered when you have it in the way you're
hoping to have it. It creates a feedback loop. Reaper responds, by default, by disabling the audio so you don't potentially blow your ears off
I might be exaggerating.. I'm not sure! Anyways, the gist of whats happening when you do it how you want to is that your drum track routes MIDI/audio to the individual parts, and those parts route audio back to the MIDI track, creating a loop of the audio being sent from the midi track, to the parts, back to the MIDI track, and back to the parts, etc... I don't know if you can just disable the AUDIO in the routing from MIDI track to the individual parts and have it work, but I think others have said that still causes Reaper to shut down the audio. This is where turning off the setting for disabling audio feedback loop may work in combination with that - I'm not sure, but beware of opening the door for accidental feedback loops elsewhere.
Anywho, I do it the way it sounds like you consider counterintuitive, but it works, and you can still tuck things away as necessary, imo.
--> Drums Folder
-----> Drums MIDI
-----> Kick Track
-----> Snare Track
-----> Hats Track
You put all your MIDI on that one labeled Drums MIDI Folder. When you create a new track from some VSTi that sets up the routing for you, it will put everything all on the same level. Just create a new folder, and drag everything into that folder and it should work. What I presume you're doing is dragging all those individual parts under the MIDI folder, and that's where you get into trouble with the audio disabling. You can turn off the option to disable audio on a feedback loop, but I don't think you really want to do that.
It does sound like you don't like doing it that way, but that's the best way I've found to do it without fighting with routing and stuff.