Popping the case is the ONLY fix. You are right, you won't get the stickies underneath otherwise.

Warm soapy water won't hurt anything but the pc board, and even that might survive it. Sweet, sticky (organic stuff) on the PC board will likely "grow" even if you try to remove it - may not be reliable in the long term. My guess is that the membrane for the buttons will save the rest of the innards.

We serviced stereos for many years and actually had some pieces come in from a flood. The Napa River in northern California floods periodically. This family said they had taken all their electronics upstairs when the waters began to rise. The receiver, reel to reel, and VCR got soaked in muddy water. The speakers survived this flood to the second floor because they had been put on a couch and the couch floated!

The guy gave me the VCR and receiver. For grins, I put them through my dishwasher. The VCR had grit in the bearings of the motors and could not be resuscitated. The receiver cleaned up really well. I sprayed the controls and switches with silicone after I was done and gave it to my mom and dad. It worked in their house for 10 or 12 years, then I sold it to someone else. So the water and soap worked OK there.

The sweet stuff seems to get stuck under components and jumper wires and never fully gets cleaned out. Ends up growing back in a green mold after a couple of months.

I think you will be OK.