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With the help of some community members in chat (thanks!), I was able to solve this and learned stuff in the process.

I'm not the first person to run into this issue; there is a [Gnome extension](https://extensions.gnome.org/extension/4338/allow-locked-remote-desktop/) that fixes it.  I didn't know Gnome *had* extensions.  To deploy this, first install the Gnome extension manager:

```bash
sudo apt-get install gnome-shell-extension-manager
```

This adds Extension Manager under "Show Apps".  Open that, choose the "Browse" tab, and search for "Allow Locked Remote Desktop".  After I installed and enabled this extension, remote connections behave as I expected: the remote client connects and I can type a password to unlock.

I am aware that Gnome is not the only UI/window manager available on Linux.  At the time I asked the question I didn't know this was Gnome, the default.  I don't know if the problem is specific to Gnome or if it's more general and this is only a partial answer.