Hello jamiedck,
Thanks for asking the Apple Support Communities about Safari.
What you are experiencing sounds like the way Safari was designed. When you hit that red button, you are closing the window, and all the tabs. This creates a new Safari session.
This is why the setting works only when you Quit Safari, it has those last session links to pull up.
Another option would be to create bookmarks, and then reopen them with one click. this article has the info: Use tabs for webpages in Safari on Mac - Apple Support
Once you have your bookmark list created, secondary click on the bookmark title, and change your option from Open in New Tabs to Automatically Replace Tabs. Please be aware, this will replace the open tabs, with the bookmarked pages. If you do not have open pages, this will open them all. Thanks and have a wonderful day.