Manhattan Mini does provide a free storage taxi, which I presume you got into and moved your stuff. We're trying to make it so you don't have to ever visit the storage unit again. So the van got you there, but you're still spending that beautiful saturday inside the concrete storage prison climbing over boxes to find that one thing you're looking for.
Also, from MMS regarding the van: Keep in mind that the Storage Taxi provides a one-way trip, not a round-trip. For example, if you need to bring more items into storage, you would book the storage taxi to help you get the stuff there; getting home, though, you'd have to rely on the subway, a cab, your bicycle, or your own two legs to carry you.
This is sucky.