I did a replacement test with a borrowed external flat-screen monitor. This 2nd external monitor works with Core, proving that there is VGA at the computer port when the netbook monitor function key is set to LCD + CRT or CRT only.
So, since there is VGA output, I can't imagine why the CRT external monitor wouldn't work with Core. It's some sort of compatibility issue that I don't understand.
I'll give lxrandr.tcz atry, as Onyarian suggested, but I have a feeling that there won't be a solution to this problem, and that the CRT monitor is just too old to work with Core on my computer.