Thanks guys.

After some further research, I am trying CDRoller. Well, I tested the trial version which reported that one of my failing CDs was in fact recoverable (it took a very, very long time to read). Unfortunately I need to purchase the software to actually pull put data form the CD, but, assuming it is telling the truth, it may be worth the expense ($ 40 for data I somewhat care about).
I am now testing the other unreadable CDs before making a decision.

I would guess there's some free alternative out there, actually.

CD readers: back in the days, I actually had few SCSI plextor readers, which had a stellar reputation. Using them now is not that trivial. I'd have to dig out or buy a SCSI controller for those.