Hey, Camilla! Trello is a great tool indeed, and I love it both at work and to keep my personal life and projects in check!
You can deal with the visible archive issue by having a general “Done” column and filter it by client when needed (keywords are very welcome here).
The way you’re currently dealing with your list of clients made me believe that the Hello Epics power-up would be a great addition to your workflow: nest individual tasks in each client’s card and you’ll easily keep track of what you guys have to do for each of them. It has a small monthly fee, but I believe it’s well worth its asking price. Hope it helps!