The timing is not a problem, at least not that I can determine. The hook does pick up the thread like it's supposed to. It's what happens after that operation that might shed some light on it. I did solve one problem that might have affected the timing, as well as shredding thread. Upon inspection of the plastic bobbin case, I discovered a couple of scratches, possibly made by a screwdriver, on the outside of the case in the path where the thread would travel around it. My trusty nail file made short work of those, and now it does sew multiple layers even on the center needle position. However, it is still skipping stitches and sews when it wants to.
I'm considering taking it in to the shop anyway for a good cleaning and oiling, and hopefully that will make it work better. Might also need some fine tuning which I don't have the tools for.