I would have thought a site might typically get uploaded in stages, over time (whether or not it's actually 'live' from a public point of view). Sometimes just a page at a time, perhaps with a minor alteration to update an existing page.
I'm sure you're right. I never work directly on live sites
I don't understand what you mean. What's being broken? Given you have a local server/database, the two sites are quite independent. I don't know what was wrong with this arrangement. To mix the local and remote causes the issues/confusion I mentioned. I would have thought it makes it more difficult to test the sites or individual pages before uploading to the remote server.
I don't like this idea. It seems related to the 'Publish' button which I've never used and am sure I never shall. Apart from sites developing, and being uploaded to the live server, over time, updates and changes will inevitably be made. I can't see that I would ever want to 'hit the publish button' in any circumstance. If that's how some people work, that's fine, but I don't know why it has meant changing/breaking what I thought were some very good features. It didn't occur to me that there were any issues in the way it was set up before.