Further details: the old disk is definitely bad. If I connect it to another computer with a USB/SATA adapter, the disk gets far enough that I can see "Macintosh HD" but if I try to actually open it, it just sits and spins for a minute before responding. I did "repair disk" on it, but this didn't change anything. It still runs slowly, if at all.

This suggests that the bad disk is the fundamental problem I need to fix, right? Not so fast. I now have the new hard drive, unformatted, installed in the laptop along with a USB boot stick. The machine noticed the boot stick and let me select it, but after that it ran for a few seconds only to display the dreaded "circle with a slash" rather than the Mac logo. This suggests that something else is wrong. Perhaps.

Again, advice is welcome here.