I'd say it would be more likely that USB external storage will still be allowed, but the SNY SSD carts will be close to the price of an equivalent external SSD, so you might as well spend slightly extra and just get the cart. That's what SNY would be hoping for anyway, since you would almost definitely want more storage.
This would also mean you're talking like a 500GB-1000GB SSD on board, with an SSD expansion port. If you add, say, a $100, 1TB cart, you now have 1.5TB-2TB of fast storage. Or you can buy an external SSD/HDD for less, but you have to transfer games to the on board SSD to play them, depending on how easy and quick that is to do.
Would you rather spend a little more money with no wait times, or would you rather spend less (or more but getting more HDD storage space) and wait, or spend time transferring game files when you have time so you don't have to wait later?
It would be a total disaster. Console is something you plug a game and play, not managing game files between inactive storage(slow HDD) and active storage(SSD).