If I'm understanding the problem correctly, the problem is that the voltage programming is not working, while the readback on the display matches the actual output voltage. I'm confused why you'd consider the ADC to be an issue?
The DAC is multiplexed, so you would need an oscilloscope to check its output. I would start by measuring after the multiplexer, looking at the schematic, the outputs of U31 (CV_REF, CC_REF, etc) look like a convenient place. If those are correct, then I would look at the regulation servo, but also check that the overvoltage crowbar (CR28) is not conducting. If the outputs after the MUX are not correct, then the MUX, DAC and opto-isolators might be suspect. That's my $0.02 based on a brief look at the schematic.