It seems (am I doing something wrong?) that I cannot find a timezone by simply using the keyboard and incremental searching (that should focus on the part efter the slash). Instead I have to use the mouse to select from a long, long, long drop-down list of possible timezones. For every team member.

Yes, you are correct. You need to search by typing from start (that is how native selection in browser works sadly).

So Asia/[place] need to be typed in full.

I know it's not ideal and I plan to add a better custom selection and search soon.

This was hacked together in like an hour to solve a personal problem.