Why use admin-ajax.php when loading pages through AJAX?

You are mixing things up a bit here. What you are currently doing is adding an optional layer on top of your website. For users with JavaScript you make the page load look different, as you use custom events/effects, instead of relying on the browser.

admin-ajax.php is relevant, when you want to access specific methods via AJAX. As described in the codex, you can create custom functionality with this.

To sum it up:

  • Your page transition script is an optional addon on top of WordPress. The site will properly function without it
  • When writing a plugin and you want to do AJAX calls, use either admin-ajax.php or the REST endpoints. If those are removed, the plugin will stop functioning as expected.