pick random name from a column when triggered, and write name and timestamp to a log

I want to allow a user to copy a list of names into a column in a google sheet, then click a checkbox or button to generate a randomly picked name. I want that name to be logged by adding it to a row of the same (or a separate) sheet along with the timestamp (in the next cell in that same row). Then I want to clear the picked name and allow the user to pick another name which would be added to the log on the next blank row.

My current sheet is at https://docs.google.com/spreadsheets/d/1EAMXXkG8nNS6lgdAZIZ6qKYo_yxsVJRllMj1B26UMdM/edit?usp=sharing