Is there a car stereo that will let me pop my CD in, save all the songs into its hard drive, then let me play them without the CD?

No, but do look around.

If you rip the CD, convert to mp3 files, load them onto a media player and patch that to your stereo through the aux input, there you'll have it. I burned more than 8 hours of mp3 content on one CD to play on my car's CD player.

Brennan audio used to do something like that

You can use the rip function in windows media player or Itunes to save all the CD format songs as MP3's on a flash drive or your cellphone if you have bluetooth to play them back or use the USB port if you have one on the stereo. The DVD player in my car will even play the MP3's off the DVD so I have several hundred songs on a DVD I can play that way right off the DVD player in my truck that's in the head deck for the rear seat DVD player.

Most modern car entertainment systems do that.

My wife has a Honda - it does that.

My friend's Mercedes does the same, so does my daughter's VW.

I don't drive so I can't comment for any other car.