I fixed it with turning off/on Use iPhone widgets switch in System Settings > Desktop & Dock > Widgets.
Turn it off, wait a minute, and then turn it back on again.
During this process, all of the iPhone widgets disappear from my desktop and then reappear and start to show what they are supposed to show.
Of course, not all of the iOS widgets are ready to be used on MacOS. Some of them that are capable of interacting with the user can't do so on MacOS. Others, like Spotify, don't even show up in the widget chooser. For example, I can view my note list on the Simplenote widget, but I can't edit it without the iPhone. The Google Search widget also does not work for this reason.
Some widgets you add on MacOS that weren't previously added on iPhone won't work until you give them permission: open your iPhone, wait for a prompt, and confirm.