Success! It took me a while to get the gaskets and tools that I needed, but I finally got it running.
I pulled the cam and found that the compression release had stuck in the "run" position (not sure what to call it, but it wasn't springing back to where it should be for starting the mower). I cleaned its hinge really well, and bent the spring out a little bit to give it a little more oomph, and after that it started moving very well. I put it all together again (new sump gasket, new head gasket, correct torques on everything, etc.) and mowed my grass tonight. We'll see if that is just a temporary fix and if I'll have to get a new cam eventually (the service manual hints that the compression release isn't exactly a serviceable part, and if it isn't working, you should get a new cam), but for now I'm happy.
Thanks all for the help and advice.