How to manufacture chip on board?
A custom chip for 10k units is unlikely to be cheap, as the masks will almost certainly cost more than $1k, which means your chip cannot possibly be made for $0.10. The reason the one you've found costs $0.10 is because they sell in a much greater volume (100k-millions), so the cost of masks and setup for a production run is spread over many more units.

That being said, a chip like the Silego GreenPAK might be a good option for you, as it's a small and extremely cheap (the cheapest on mouser is SLG7NT4375 at $0.195 each for 1k) CPLD. Assuming your design fits and doesn't need too much analog functionality, this should work for your use case.

A micro controller isn't necessarily overkill, and may also be pretty cheap. For instance, the lowest cost mcu on digikey is the ATTINY5-MAHR at $0.14 each for 1k. Additionally, if you're willing to navigate the markets in asia, prices can potentially be even lower.