Currently, when I click on that menu on the front page, it takes me, for example, to
https://smittenkitchen.com/2015/03/wild-mushroom-pate/ (which is a post, no page)
next time to
https://smittenkitchen.com/2010/04/cabbage-and-lime-salad-with-roasted-peanuts/ (which again is a post, no page)
So I cannot see that your current solution reloads a post “on the same page”. It’s always a different post with its own URL, and no page involved (of course, the theme/design is the same for all posts). Here is an example for a page: https://smittenkitchen.com/books/ (there’s no date in the URL).
Or do you mean that it sometimes opens a new tab in the browser? When I try, it’s always in the same tab.
On your front/home page there is a list of posts. Do you want to reshuffle their order randomly, but still show posts as a list? If it’s that what you need, this plugin cannot help. This link could give a hint how to do it: https://www.wpbeginner.com/wp-tutorials/how-to-display-random-posts-in-wordpress/
If this is not what you need, could you paste two links, one of a “new post page” and one of “a different post on the same page”? Seeing a good and a bad example might help me understand.