Packing a wet tent away should be no more than a minor inconvenience. Whenever I've had to do it, it's always been possible to get it dry again quickly once the rain stops. It generally only needs a really good shake, then half an hour's blasting with a dry wind. I can't remember when I last had to put a wet tent up, and I'd always be prepared to spend up to an hour getting a tent dry during the day, rather than unwrap it wet in the evening.
For the record, the last two times my tent has been packed away wet has been, of all places, the dry and arid Gran Canaria last year! Maybe I was just incredibly unlucky, but on two mornings in the middle of nowhere, after completely dry nights, it absolutely hammered down with rain only a minute after I'd started pulling the pegs out!
Same year... ten weeks in Iceland... and my tent never got wet!