The PDF is showing it needs a special driver in order to read the card as per below:
Driver Needed: ATDOSHC.SYS
Must be installed in CONFIG.SYS as:
DEVICE=ATDOSHC.SYS
In order for the hardcard to work in a given system, the user must assign a unique IRQ and a BIOS address.
The default for this card is IRQ 11, BIOS address C8000-C9FFF.
SW8 is reserved for IRQ 14 (IDE MODE).
However, since the drive doesn't have any other drive, I don't have a config.sys file to edit, nor do I have the driver file. The jumpers were all over the place so I reset them back to the default shown in the PDF and now the system boots to a normal "Insert System Disk" or something along those lines. At least the NO ROM PRESENT" error is gone and it actually tries to boot from a floppy. However, I can't get it to boot from a floppy. I created a DOS 6.22 disk and tested it on a laptop I have and the laptop boots perfectly fine with it. This 386 seems to only boot from A, but B and whomever configured this 386 had the drives reversed (1.2MB 5.25") was A & (1.44MB 3.5") was set as B and the system would only try to boot from A so I changed the cable around and reset the BIOS and it attempts to read the DOS boot disk, but just gives a drive failure error. I tried a spare 3.5" 1.44MB drive I had laying around and same result. I'm pretty sure the hard drive is OK. It's spinning, does a normal head seek at boot, no click of death, etc. It's acting & sounding like any other Quantum drive I've owned. I think if I was able to get the system to boot as well as load the driver for the card, it would work perfectly fine.
For the life of me, I simply can't get this 386SX to do anything. I dunno. I'm almost tempted to gut this 386 system and trash it and build a Socket 7 System out of the case. I don't even know what I'd do with a 386SX-20. The 486 Zeos has a DX-33 in it which isn't swappable so I bought a new motherboard with a DX2-66 included. I still don't see a point in having a 486/33 & 486/66 though. I dunno, getting a bit annoyed at these systems at the moment.