Here's an idea you can try: Open a view in Tableau Online, click the "share" icon to the top right of the view, and then copy the URL under "Link" (you'll see fields for Embed and Link). Then open a new tab and paste in that URL. You'll see that the view is presented in a very clean way. I have to admit I'm not 100% sure if this is the same as "Presentation Mode" but let me know if it meets your needs.
Also, thanks for the feedback on URLs changing. If you publish a workbook from Tableau Desktop to Tableau Online or Tableau Server, and then go back to Desktop to make changes, and republish with the same name, it should leave the URL unchanged.
Hoping this helps,
I have found that the URL changes every time you republish to Server.
You can edit the URL and we are testing this today, to ensure that it take users to the right place, however it is not the URL provided in the Share option on Server. That URL will go stale if you make edits to the workbook and republish. Whoever you sent that Share link to will no longer be directed to correct place on server and will receive an error message.
I will try your suggestion above. Thanks. Angie