Siri stopped working after iOS 16 update

I checked the Siri settings right away. They were all unchanged from before the iOS 16 update, yet Siri no longer worked.

As I noted earlier, when I made a request, Siri waited a few seconds, then said, “On it.” Then after a few more seconds, said something else about still working on it. Then the process failed completely, displaying (not speaking) an error message. I got the same responses whether I asked Siri to make a call or to tell me the time of day. Siri was broken.

I finally resolved the issue by making an iCloud backup, then doing a complete reset, wiping everything as though I were switching phones. After going through THAT ordeal (which included also resetting my Apple Watch), Siri began to work normally again. And it still works normally days later.