Have you tried connecting your optical drive(s) to the motherboard's header and letting an add-in card control the hard drive?

That's how my system is configured and it seems to work fine, though it's IDE only.

Also, my search for Windows ME drivers for those cards didn't turn up anything from sites I trust. Perhaps that's part of the problem?

Couple things to try:
first, update the bios'es to each card.
Make sure your motherboard's bios is also updated.

I personally use a card very similar to the TX2300 (i use the TX4310). I personally do not install any drivers, because they seem to cause more issues than solve in my use case. The card is smart enough to do everything automatically. the 2300 should be very similar. I find it VERY strange the cards you listed don't see the CD or DVD drives. UNLESS you are doing something silly like using a SATA to IDE converter. Never use converters unless there is zero other option. Another pitfall are crap cables. I've bought NEW cables from the usual skumbags and even the new ones can be faulty.

I also found its imperative to disable the on-board IDE. Really need to free up those IRQs for the card. Neither Winblows or the BIOS are smart enough to do it. All your drives need to be on the sata card OR the onboard ide. There has not been a case for me where having both played nice together. Late Pentium 3 and newer boards are fine though.

food for thought, the CD drive might also be at fault. I'd look into a new cheap drive. Another plus for the tx4310 is large drive support. I had zero issue using up to 137gb of my 320gb drive I use. The controller can see more, but i'm at the limitation of FAT32 at this point.

luckybob wrote on 2020-04-07, 23:53:

Couple things to try: first, update the bios'es to each card. Make sure your motherboard's bios is also updated. […]
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Couple things to try:
first, update the bios'es to each card.
Make sure your motherboard's bios is also updated.

I personally use a card very similar to the TX2300 (i use the TX4310). I personally do not install any drivers, because they seem to cause more issues than solve in my use case. The card is smart enough to do everything automatically. the 2300 should be very similar. I find it VERY strange the cards you listed don't see the CD or DVD drives. UNLESS you are doing something silly like using a SATA to IDE converter. Never use converters unless there is zero other option. Another pitfall are crap cables. I've bought NEW cables from the usual skumbags and even the new ones can be faulty.

I also found its imperative to disable the on-board IDE. Really need to free up those IRQs for the card. Neither Winblows or the BIOS are smart enough to do it. All your drives need to be on the sata card OR the onboard ide. There has not been a case for me where having both played nice together. Late Pentium 3 and newer boards are fine though.

food for thought, the CD drive might also be at fault. I'd look into a new cheap drive. Another plus for the tx4310 is large drive support. I had zero issue using up to 137gb of my 320gb drive I use. The controller can see more, but i'm at the limitation of FAT32 at this point.

Thanks. I was thinking to update the bios of the TX2300; anyway the TX2300 I tried seems to see only hard disk even in others configs I tried in the past. I'll check with different dvd drives as suggested but still I don't think these will boot anyway from it cause it seems like the second or others sata ports are there only for RAID modes and not as secondary ones.
If I use it only for the hard drive I still have problems with the on board IDE that seems to have DMA enabled but working as bad as without it like something is wrong in the IRQ/DMA configs. Windows ME has most of their drivers so it should not be a drivers issue with these old i440BX chipsets.