This is one of the reasons I like my single-wall silnylon tarptent. No worries about setting it up or packing it wet, and it easily fits in a pannier. There are drawbacks to a single-wall tent, so I need to harp on the positives when I can!
Back when I traveled heavier and carried a tent with fly, I don't remember worrying about the tent getting wet. That's kind of what it's supposed to do. I'd dry it during sun breaks if I got a chance.