Something that Chrono Cross Disc 1 needs have enabled is the "Odd/Even Bit Hack" in the video options. Apparently Disc 2 doesn't need it, but Disc 1 does to help alleviate these black screens.
Go to Video > Configure > "Misc" section > Check "Special Game Fixes" > Press the "..." button > Select "Odd/Even Bit Hack"
Even though it says it's for ePSXe versions 1.5.2 and older, enable it. Doing that fixed those problems for me.