I love platform beds it is all we have ever had. IMO a box spring is the biggest waste of money as all it is is a box to support the mattress and 1/2 the time costs almost as much as the mattress. As for a style that allows under bed storage that will require you going shopping. I like this kind of storage under a platform bed for extra bedding you don’t use all the time but it will depend on what you want to store under the bed. If you get a custom made oversized coverlet then no need to have a bedskirt and it eliminates the need to tuck the bedding in .There are many platform beds with actual storge drawers under them

Often, platform beds are lower than traditional ones so you'll need to figure out the amount of space you need for under bed storage. Forgoing a bedskirt means the under bed storage will be visible. Another option is a bed with built-in storage.

Bed frame with storage drawers would fit your preferences.

Don't know what your price point or style is, but Pottery Barn makes a platform Captain's Bed that has storage baskets or drawers underneath.