I’m not suggesting this is the final solution, but this is something you could try that might help to solve your problem.
As it happens I have a Realtek soundcard on my computer. I don’t normally use it because I use an ASIO setting with my Line 6 UX1 audio interface.
However, by reverting to Directsound, I just managed to get a sound out of Reaper using these settings.
As a test maybe you could see if this works for you.
Pete