I ran SQL queries to fix the parent category (reset it to zero) for the problem categories as well as re-setting the term_group to zero in wp_terms, and that helped restore those missing categories when listing categories on my local copy.
But it didn't solve the problem of what appears on the category archive.
There is still something fundamentally wrong with your data, though.
In the screenshots you can see that when listing posts it reports that the latest post "About Judy Manherz" has the categories Article, Realtor, and Team. But when you edit the actual post in the second screenshot you can see that it reports that the post has no categories assigned, and not only that but there are no existing categories to choose from.
And this happens with all plugins disabled and the twentytwenty theme.
So while there are underlying problems with your data I think it is expected that anything dependent on categories will not necessarily work as intended.
When you make a custom WordPress Archive for categories, the post types included in the query will by default be any post type that categories are assigned to. Using the block editor to create the archive currently doesn't let you change this, but the classic editor does.
With the Query Monitor plugin I was able to see that visiting site.com/category/article it was only including the property post type in the query, and not built-in Posts.
So on your local site I deactivated the custom archive for categories and created a new one with the classic editor.
There I can see that by default it only wants to include Properties (screenshot), which must be connected to the underlying problems with the data. So I can go ahead and change that so that it includes both Posts and Properties, and then the archive works as intended on the front-end, returning both posts and properties.
But this means that you need to create and design the archive with the classic editor and not with blocks (go to Toolset > Settings and you can choose which editor to use when making a new archive).
If the underlying problems with the data are resolved I would expect it to be possible to create a new archive and for the assigned post types to all be included, and if not I could investigate further, but the data problems would need addressing first.