My plan is to cut it into strips roughly 2" wide to pack into survival/first aid kits.
Why not reroll it on a card? Well, I do that sometimes but I've noticed that with many brands this degrades the tape, and the re-rolled product often doesn't stick as well as it does new off the roll.
I've never been a fan of re-rolling duct tape, for the reasons you note. However, REI sells small rolls of SOL brand duct tape. They are tight little rolls that pack nicely. I keep a few around, in my field repair kit and what not. They have come in very handy on several occasions. The only problem I've found is that it is tough to get the last bit unrolled.
SOL duct tapeRecently I noticed that Lowe's sells Gorilla Tape in a small (1" X 30 ft) roll. It is a bit larger than the duct tape rolls noted above, but still a handy size.