I am having the exact same issue. And it is not the cards not getting mounted. I've manually mounted a memory saved it, and looked at the SSD and it has not changed. Yeah currently the only way to save progress on the psx core is using save states, which sucks. I should also point out that I have a system that has worked for years, I have old saves that were working fine in a previous update but now can no longer be saved (updated).
I went ahead and did some testing. I switched the core all the way back to PSX_20220511.rbf and I was still having the issue. Before I saw this message I thought the issue might be related to the game. So I tested another game I have and it worked fine.
Game with issue is:
Arc the Lad I
Game that worked fine: