I could replicate the issue on a clean WordPress installation, on all default themes. What was interesting is that the feature would sometimes appear, and then disappear again.
Having a look at the console, I noticed that it didn’t show up when some requests where failing, which I believe are internal REST API calls failing because of the security system (this is mentioned in the following article):
One of the requests failing is wp-json/wp/v2/taxonomies, the request for taxonomies (tags and categories) which explains why you cannot set them.
One solution I’ve tried out is to install Classic Editor and use it alongside Gutenberg, creating the main content in the block editor, and using Classic to set the fields you’re otherwise unable to in the block editor. It’s not ideal, but it should work, as Classic doesn’t seem to use the REST API.
If you have a custom domain, you could also set up Cloudflare, which bypasses the security system’s limitation.