Most people believe that Taipei tap water isn’t safe to drink. My suspicion is that the water that arrives in the building is perfectly fine to drink, but most buildings then have holding tanks, and those tanks can be suspect.
We used to get big bottles of water delivered, but a few years ago had some kind of filtration system installed under our sink. I’m not sure if a Britta filter would really be good enough.
I’m perfectly happy drinking boiled tap water and have never had any problems, but my wife prefers to use purified for boiling as well. I think many, perhaps most, families in Taiwan just boil tap water and don’t treat it otherwise.
I certainly don’t worry about brushing my teeth or showering with Taipei tap water.