A bit late to the game, but I can report that the Quad SATA HAT works fine on the RPi 5. The 2xUSB to 2xUSB connector is a bit of a tight fit. I’m not using the original case, just the bare board. The RPi 5 is fitted with the standard heatsink/fan.
To boot from it I’ve enabled it in the RPi’s EEPROM settings as per @Craigp’s post.