You still might lose a rod but that gives you a few more seconds to react without having to bring more stuff on the ice.
That said,I never wander away to pee on the ice. I kneel down and pretend to be doing something with one hand while holding "myself" up off the ice with the other....I make a point of going light on the beverages out there to keep pee-breaks to a minimum.
They are almost like a tipdown, just don't produce any slack line. You place your rod in the slot so it balances, set depth, and leave it. Then watch for the rod tip to start dropping, pick it up and set the hook in one smooth motion.
They fold flat and you can fit a LOT of them in a 5-gallon pail.
Seriously though, you can reel up out of reach of fish while you cannot pay attention. Consider a tip-down rig that may give you a little more time. I would not mess with the loose drag idea cause it'll screw you up on your hook sets.
Don't feel bad, I have lost a few jig sticks along the way (over 40 years fishing), as I'm sure many others also have. Sometimes no matter how hard you try this can happen while having good action... you're just busy playing a fish on pole A, while seeing pole B disappear out the corner of your eye. This all happens while pole C never gets touched because your cat peed on it in your basement last night. Hahaha, just having fun.