Is it normal operation for web editor to delete a web part on a page and have it be reflected immedi

This is kind of bad because while editing a webpage, when the editor wants to remove and add different web parts the resulting live webpage will look funky to the public until they are done with their edits. If it is something that will take a long time, the customer could be prevented from getting some of the information on that page due to layout issues. I wish it wasn't this way but will just have to educate our editors.

Webparts are part of a page's structure and should not hold content specifically. It's designed to work this way. Many CMS solutions do the same thing. Sounds like you may need to better understand how a template works.

It defeats the approval process for workflows.

Say we have an approver (manager) and a web editor (worker). Manager tells worker to change a certain webpage, remove the old calendar content and add new calendar content with a different style. So worker checks out the page in the CMS and proceeds to delete the webparts that make up the old calendar, and begis working ont he new calendar layout.

Unbeknownst to them, the current webpage suddenly has NO calendar data because when the web parts that hold that calendar data were deleted from the checked-out webpage, they immediately disappeared off the live website also! The worker steadily rebuilds the webpage and after a couple of hours is ready to have their web work reviewed by their manager. So they Save and SubmitForApproval.

An hour later the manager reads the CMS approval email, takes a look, sees it looks good, and approves the changes, at which point the CMS publishes the content changes out to the live webpage.

Now the live webpage looks good once again, but has been ugly for a few hours. This doesn't sound like a good way to do webpage editing. Better would have been to only remove the deleted web part locally so the Preview option shows it gone but the live site still shows it there, until the page has been published.

I would recommend signing not only your editors for Kentico trainings to learn how things work and maybe instead of web parts use widgets for the "content" parts you want to have under workflow. It is also explained in the documentation, in a big yellow warning part of the page that the changes are reflected right ahead.
The Design tab and in general access to page template requires a special permission - this was added so only trusted and educated users have access to this as by editing the page template there can be severe damage done or data can be lost. I would recommend to get your editors on some training to learn how to work with Kentico to prevent bigger issues or even introducing a security issue in the future.