I had the same problem.
My browser was set to block pop-ups. You need to allow pop-ups.
On my browser the option to allow pop-ups appeared at the top of the page each time I tried to use the 90-day system, but I hadn't noticed it.
Visa on arrival and Visa exempt are two different things.
Taiwanese do not get visa exempt, they have to apply for visa on arrival.
Edit to add: Visa on arrival is normally 15 days.
Taiwan (Republic of China) is not recognised by Thailand. There are only about a dozen countries that do.
https://worldpopulationreview.com/country-rankings/countries-that-recognize-taiwan