Hi Nikki,
If they aren't responding to your requests for common courtesy then you should absolutely take charge of the storage space. I think your "abandoned property" idea is a great way to give them time to organize themselves before you act. I would actually encourage the same thing for the laundry room. I once forgot an entire load of laundry at a laundry mat and when I went back in the morning they pointed to a sign that said "Any items left behind after hours will be thrown away" I was definitely disappointed and a little irritated but I only had myself to blame. I mean, I left an entire load of laundry behind!
If you approach the subject from a respectful standpoint, those who are still careless with their items also wouldn't have anyone to blame but themselves. Especially when they have been made aware of the changes. So maybe instead of throwing things every night at 11pm you hold them for 24 hours where they can pick them up for a fee or something. I know it sounds silly to treat adults like children by setting consequences to bad behavior, but we shouldn't be in the business of accommodating poor tenants at the expense of a good one.