The pickup for “friends” and “family” means anyone in reality. Most of the pickup cars are actually driver services, meaning not taxis but companies that offer airport pickup and dropoff services. Ie Black VW transporter vans with a driver. We use this service as well (we get free pickup / drop off as a credit card bonus, for Taichung makes it easy). Driver had no problem also stopping at seven for some shopping and bathroom brakes on the way.
You can also walk to the car park if you want (since you can now drive your self also) but the pickup area is very convenient. Btw you can also smoke without a mask at the pickup area together with the airport security guards and employees in white bunny suits that help people go to the right place. Just walk all the way to the end, its seems to be the dedicated employee smoking area. Very useful first to spray alcohol on the floors for an hour and then go have a smoke and chat with the quarantine arrivals. The whole thing is so useless.
I am not sure about taxi, but you can certainly do it in your own car. I just did it last week. My wife and daughter were returning to Taiwan. I drove my car to the airport, picked them up and brought them home. No problems at all.
P.S. Nobody cares anymore. I picked up my wife and daughter and drove them to Kaohsiung. Instead of 4 hours we took 6 hours. I made multiple rest stops for food etc.
I dropped them home and went to stay at a cheap hotel for 3 days. No one ever came to check on them. If I had stayed home with them, no one would have known. At this point, it all seems like meaningless exercise.
Then why are we still making inbound travelers quarantine? It’s not about whether the virus has breached the border. It’s about the quantity of the virus.
According to this graphic, circulating in another COVID post here, public transportation is allowed during self health management, provided you have no symptoms.
I came back yesterday. Hardly anyone at the airport, plenty of taxis, capped at 1000NTD. There is the free bus to various points (linked to on the airport website) but you have to have someone you know pick you up there, you can’t just jump on public transport when they drop you off.