I want to say thanks to everyone who helped me with this. I eventually chose to go with the Fractal XL with an additional drive cage and a 4 x 2.5" SSD to 5.25" bay adapter. I realized that if I'm going to have up to 16 drives, a RAID 6 is not enough to ensure data security - I needed a RAID 60. Since with RAID 60 you need an even number of HDDs, having a case that supports 15 HDDs is no more useful than one that supports 14 HDDs. The Fractal XL is the only case I could find that would support 16 HDDs + 4 SSDs without jerry-rigging a mount for the 16th HDD, the 4 SSDs, or both.
The extra drive cage (I bought the "top' one) required a bit of modification to get to fit. In the end it didn't matter whether I bought the top or bottom cage since there were no mounting points on the bottom of the case for either one when the other two cages were in place. To mount the extra drive cage, I removed the plastic "feet" from the bottom of the cage, which left four holes in the bottom of the cage. I then fed screws through the holes where the "feet" were and then through the honeycomb fan grille, raising the cage off the bottom of the case using a lot of washers.
In the end I'm happy with the result. It's significantly quieter than the old system due to the sound deadening and quieter standard fans, which is a nice plus. I also as a precaution retired the 10 year old Silverstone power supply for a Seasonic S12G-750, and that's also much quieter than the old supply.
Thanks again everyone!