Yes, choosing a prepaid rate does not guarantee a vehicle. You are basically in the same boat as someone who booked a pay later rate. That said, booking a Minivan or a Fullsize SUV is not like booking a Range Rover or something. Even on a holiday weekend, the rental companies usually have enough of these vehicles on-hand. Minivans are somewhat more prevalent than Fullsize SUVs but if you need room for 6, either would work and if you don't care about the vehicle type, just book what's cheapest as one can always be substituted for the other if they run out of what you booked.

While you could book a backup reservation with another company, that sort of exacerbates the problem of vehicle availability/no shows/etc. so we tend not to recommend that. What we do recommend is to book with one of the 3 major companies (comprising 8 or so brands) and stay away from the discount companies as they don't have the breadth of inventory to pull from when things get tight.