This had nothing to do with Google Sheets itself, but rather an extension I use called Don’t Fuck with Paste. This extension disables copy and paste blocking.  The extension had a major upgrade a couple months ago that enabled the extension only for sites added to a blacklist.  The blacklist supports regular expressions to match URLs.  Unfortunately, a blank expression matches everything and somehow a blank RegEx was in the blacklist–I don’t know if this is the default for the extension or if I did it by mistake and I haven’t tried to find out.  You can tell if the extension is active for a site if the extension’s icon is blue rather than grey.  You can tell if a blank RegEx is in the list by clicking on the icon and looking if there’s a cancel button next to a blank line:

instead of this (the URL displayed will be the one displayed in the current browser tab, but the important thing is that there’s no X next to it):

Clicking the X removed the blank RegEx and turned the extension off for all sites by default, thus fixing Google Sheets.

On finding extension problems

When something goofy is broken, the first thing to do is to browse to the site with an Incognito window, which turns off all extensions.  If the site works in Incognito mode, the problem is due to an extension.  The next step is to figure out which extension is breaking things.  If you don’t have an idea of which extension might be the problem, a binary search is the fastest way:  Disable the first half of all extensions, reload the page and see if it works.  If not, re-enable the first half, then disable the second half.  Keep splitting the groups in half until you find the extension at fault.