Oops. Looks like I fundamentally do not understand how networking works with mods, which makes sense since I learned all of this in 3 days from frowning at the source code for other mods and sort of shrugged off all that "isRemote" stuff...

Time to figure that all out!

Okay, scratch that. I actually cannot reproduce the issue I was having, which means I almost certainly did something silly and should be ignored.