In the hope that this might be helpful, I tried another SATA PCIe card (this one also has a sil3132CNU chip and allows you to use SATA or eSATA SSDs depending on the jumper settings), this model:

https://sitecom.manymanuals.com/interface-cards-adapters/xc-043/datasheet-794

Unfortunately, the result is identical to that of the other card: MorphOS fails to boot properly (no video output), and after several seconds, the PowerMac G5’s fans can be heard running at full speed. In contrast, OS X Leopard boots perfectly and detects the SATA PCIe card, just as it does with the RTL8168 PCIe Ethernet card and the ATI Radeon X1950 Pro PCIe graphics card.
So I’m afraid I’ll have to put aside the idea of connecting a SATA2 SSD to the PowerMac G5 for now.
If there's anything else I can do, I'm here to help.