Easy default search replacement

From what I can glean, this would be amazing on several of our education/informational websites.

For example, on https://www.kobejet.com/, we currently use the default Olivero search bar with the search autocomplete module (screenshot attached), and it works pretty well.

But obviously this would be a leap even beyond that in helping teachers, etc. find lessons and information they need faster.

So here's the question - as Olivero is the default Drupal front-end theme, is there any possibility of adding an easy drop-in replacement feature for this module? Possibly appropriating the same blocks, etc. to make configuration painless?