I have a Monkey Wards riding mower from about 1985 or so. It has a 2-cylinder, 16 HP Briggs and Stratton engine that was giving me fits a few years ago. Eventually, I found the compression on the left cylinder was nearly zero. Then I got to thinking (which is something a man should not do).
The previous summer, while doing some other work, I found a mouse house similar to the one you show in your pics. I had cleaned it out thoroughly but on reflection, I realized that I had probably operated that machine for several hours with that nest in there. That means the left cylinder didn't cool properly. At one point, it had overheated so badly, it caused the exhaust valve seat to come loose in the block.
I tore the engine down and took the block to a machine shop. They used a Chevy valve seat that was about the right size, flycut the block and installed it in there for me. Not a lick of trouble since but I'm pretty religious about pulling the blower cover off a couple times a year to check for the Mouse Family reunion. I've had to pull small nests out of there several times.