You are currently in the site settings interface of a top-level site, the top-level site doesn’t have a parent site so it can't display the same navigation items as a subsite site.
You can enable this option in a subsite. If you are in a subsite, it will display as below:
No, this is not supported and if you do go forward with it, your solution will break over time as Microsoft makes changes to the DOM.
If you are using Site Pages, the link in the navigation is actually updated automatically if you rename the page. And if you delete it, it disappears. I tested it just now.
But this doesn't work for publishing pages (the link doesn't get updated and you get an "page not found" if you rename it).
I’m not seeing this behavior in my SharePoint 2019 environment, so it should not be a bug.
Also programmatically updating navigation elements is not yet supported for modern experience, so it should not be a customization error as well.
Looks like an isolated temporary issue on your side. Consider rebooting your machine and see if the issue will go away.
I’m afraid that is not possible with the OOB settings in Navigation.
The current settings only provide the ability to hide them all at once, or one by one.
Maybe you should seek solutions from custom code, if that’s something you are familiar with.
The Top Link Bar option is only available when you have the publishing features deactivated (SharePoint Server Publishing Infrastructure feature under Site Collection Features to be specific).
For sites that have publishing features turned on, you would see Navigation instead.
Reference: Customize the top link bar.