So I decided to try out a local board manufacturer (thegioiic.com). Their online order form was pretty good (with instant quote), and I could just send them the Gerber files from KiCAD.
I optimized this board to be single-sided, and decided to drill the holes myself, as I don't really need vias in this design. The battery, LED, and tilt switch mount from the verso, so the pads are fine as-is without plating the through-holes.
The board quality was consistently good overall. A few very minor cosmetic scratches (and I'm being really picky here), but nothing that would affect performance.
Also importantly, it cost about 450k VND (20$ USD) for 10 units delivered to my door, pre-cut, tinned, etc. I chose the slowest (and cheapest) service, and the boards arrived exactly 16 days later as promised.
The staff communicated via Zalo, but thankfully had very few questions (my Vietnamese is not very good). Mostly they wanted to be absolutely sure that I wanted black color boards, as the price of dying them black had gone up significantly. I took their advice and switched to green :D
This board is very simple -- I wanted to see the quality and service level firsthand before trusting them with something more complex or time-sensitive. They've passed the challenge and I'll surely use them again in the future.
I'm still not sure if I'll make a minor product out of this board someday. I rather like it, and I think it was neat to try and engineer a product that could last 100 years. So we'll see -- certainly I'll need to factor in ROHS compliance and CE certification. The latter can be expensive, but for something this simple (and that uses no high frequency signals) I can investigate the possibility of self-certifying.