The fantasy, imported from other Asian postings, is that serious care happens in another country. The fact of Taipei is that you can see a specialist on a weekday morning and be eating lunch afterwards. The barrier is not quality. It is vocabulary and the ticket machine.
Large hospitals — NTU, Cheng Kung in Tainan, Chang Gung, Mackay, Cathay — have international or English-capable counters. Smaller clinics are often better for the ordinary things: a throat, a dermatologist, a dentist who has seen a thousand night-market stomachs. The NHI card is swiped; you pay a modest co-pay; you wait.
Bring a translation of any long-term medication. Allergy notes written in English only will be treated as a puzzle. For mental health, the English-speaking list is shorter and worth asking a GP to help you find, rather than starting from a Facebook thread at midnight.
If you do not yet have NHI, you can still be seen as a self-pay patient. It is more expensive and still cheaper than the story you were told before you landed. Get on the system as soon as your status allows. That, more than any clinic list, is the advice.