My experiences with dentists here:
First, I called Lin & Partners (Tianmu branch). From the compiled list of recommendations they seemed to be the safest choice. They don't accept NHI and are probably the most expensive, but I'm fine with that as long as the quality is also fine. I didn't want to take any chances.
I was told I could make appointments in English there, so I started speaking in English when I called them. This was not a good decision, as the receptionist's English comprehension was clearly insufficient. Even though I spoke rather slowly she could not understand me, and when I said "I need a filling," she thought I needed "cleaning." Then, out of the blue, she switched to Chinese. The conversation went much smoother from then on, but if I couldn't answer her in Chinese I doubt we could continue at all.
Although I called in the morning, they did not have free slots for the whole day, so I decided to go somewhere else. It was just a single phone call so I might be wrong, but my impression was that they already had enough customers and didn't really want my business.
My second attempt was to call the Summit, which seems to be the place most often recommended on Forumosa. When I called them at the time they open (2.30 pm), they said they were full and would not make an appointment for me: I could just come and wait, but this was something I didn't want to do, so I decided to try yet another place.
This time I called the Landmark, the workplace of Dr. Hsu also recommended on Forumosa. He doesn't work there anymore, so (somewhat reluctantly) I made an appointment with Dr. Chen. He turned out to be a very good choice, fixing my long-neglected cavity really well, giving me more time than necessary and also explaining things as he went on. I got local anesthesia when I asked for it. The amount was sufficient and Dr. Chen waited approx. 10 mins for it to start working, so I didn't feel any pain at all. (This part I was worried about because many people had mentioned problems with anesthesia in other clinics.)
I went on to see Dr. Chen again later for cleaning and some miscellaneous stuff. Each time he spent about 1 hour for me, and there was no time pressure of any kind. The room is for a single patient only and there are window blinds, so the degree of privacy is higher than usually in Taiwan. (I saw pictures from another recommended clinic, I think it was Duen Ping, and there were 5 or more chairs in one large room without even any curtains between them. It is also common for dental clinics here to have large windows anyone can see through from the street.)
I can definitely recommend Dr. Chen at the Landmark. We spoke Chinese but I believe he also speaks English. Also, this clinic accepts NHI now, so the total price I paid was 150 TWD each time.
Also, I later found Dr. Hsu and in fact also made an appointment with him. He works at another clinic now (Professors'), where he specializes in prosthetics, but he was also OK to see me for the cavity. I cancelled the appointment though because it was not necessary anymore. The cost of having a cavity fixed there would be about 3500 TWD (including anesthesia but not root canals), they don't accept NHI.
Hope the above helps someone else in the future.