A fast internal SSD/HDD (used as a reliable low latency storage by developers) would forbid the use of external HDDs to store games. Or they would need to add 2 HDDs in the console: a mandatory fast and an optional slow -> costly for the manufacturer and more work for the developers.
This is one of the reasons SSD/HDD hybrid is better if the SSD and HDD are separate, then.
Seagate "SSHD" where both are inside a HDD's casing are the best known solution perhaps, it's transparent to the OS and user and is also easy for OEMs (laptop or iMac can use HDD, SSHD or SSD). It's a drop-in solution which doesn't really require anything in term of software or motherboard's firmware or OS configuration.
But for many years using a separate SSD has been available, even on consumer hardware : Intel's "SuperDuper Storage Accelerator" (I'm too lazy to search for the name lol) on Z68 or Z77, and the ZFS file system on Solaris, FreeBSD, in a lesser way Linux.
Having an M.2 PCIe slot on the motherboard would make it easiest : base option of 64 or 128GB, high end option of 512GB (adds the ability to store whole games basically, or most of their contents) or ability to put a 2TB drive or bigger in.
You can also sell a base model with no caching SSD present whatsoever. As in, most people probably are interested in the $399 version with price drop to $299 after a couple years, not in something $50 or $100 more. But then the developers have to make the game play at least decently without an SSD.
Another option is some kind of ReRAM, 3D XPoint etc. soldered on the motherboard for each console, no idea if that can get cheap enough for a small amount.. At least it should be more reliable. If you put in a small amount of flash like 32GB, would it be slow and die from exhaustion? Or with writes just fast enough to keep up with the HDD's reading bandwith, if you have good quality flash.