Most foreigners with permission to work actually have work permits (issued to their employers), not “work visas”.
When you talk about residency, keep in mind that being a resident for tax purposes is measured only by the number of days per calendar that you stay in Taiwan (at least 183 days), and being a “resident” for conventional purposes usually means having an ARC (separate from your visa).
The two don’t always go together. You can stay the whole year on visitor visas, visa exempt entries, and/or extensions but still not have an ARC and therefore not be a “resident”.
About work and boredom, some kinds of volunteer work are permitted for some foreigners, but the legal situation is still quite restrictive.
It’s even more restrictive for online work, which you basically can’t do (legally) unless you’re a permanent resident or married to a Taiwanese, which obviously you won’t be.