I've never had anything but nylon covers. I wouldn't mind trying vinyl or neoprene in the future.
My boats with smaller cockpits seem to keep the cover on while traveling, even very long trips. Larger cockpits less so.
One thing that may make a difference is that we use cheap beach balls in our cockpits under the cover. Primarily to keep the rain water from pooling on the cover, but it may help keep it on as there would be less flapping around.
As for storage, most certainly. I went to pick up a loaner kayak from my bro-in-law, who didn't cover the cockpit, and had to get two black widow spiders out of the boat before I would get in it. I've no particular fear of spiders, but that's just not going to happen.
Hope this helps,
PFD's (Life Jackets)
Kayak Deck Gear Bags
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