Visa headache... How to legally find work in Taiwan?

My plan was to enter Taiwan on a visitor’s visa, look for and find work, and then convert my visitor’s visa to a work permit/ARC in Taiwan. (BTW is there a difference between work permit, ARC, and work visa?)
TEC(R)O in the US is telling me that I have to get a work visa from them, that even if I enter Taiwan on a visitor’s visa or visa-exempt, I have to go back to the US to get a work permit from them if I want to get a job in Taiwan. Is this true?
If not, what’s the right/best way to go about this?

Yes.
Work permit explains it by the name.
If you have a work permit before you enter Taiwan, you can get a visa based on the work permit.
Arc is a residence permit. You can get it based on your work permit.

No, it is not true.

If you find a white collar job providing your work permit, you can convert your status without going out of taiwan. If you have a visitor visa, there is a restriction on which ones can be converted.

What documents do I need to prepare when applying for a resident visa for employment purpose?
https://www.boca.gov.tw/cp-181-481-c2487-2.html

Instructions for Application for Residency Status by Foreign Nationals with Visitor Visas(Serial No. 0915)
https://www.immigration.gov.tw/ct.asp?xItem=1090141&ctNode=30085&mp=2

There’s some kind of misunderstanding. What do they mean by “work visa”? Some visitor visas give you permission to work (so then you wouldn’t need a work permit), but a normal visitor visa (such as a “tourist visa”) does not.

There’s another way; why don’t you try in the other way around? If you have significant formation/experience, you can search for a job and later on, if you find someone who’s interested, try to have that employer make you a work visa and come in a completely legal context without risking being stuck in a limbo if you don’t find a job, or find one, but converting your visa or creating a new one takes too much time.

I guess a “work visa” in your post means a resident visa based on employment.