Hi @michaelxxx,
Unfortunately, the search on the header is different from search form that actually on search pages, which are usually customizable with searchform.php
The one on the header is a theme-specific search feature which has no relation to searchform.php, the one you’re referring that supported by 99% of the themes out there is searchform.php. And it’s the form on search pages, not on header bar which again, is usually theme-specific.
As for reference of which is search page and search form, then please check https://codex.wordpress.org/Creating_a_Search_Page along with searchform.php https://developer.wordpress.org/reference/functions/get_search_form/ . Is completely different from the one on header.
You can only turn off the header’s search bar, and integrate your own search to the header. For example, you can check this https://www.wpbeginner.com/plugins/how-to-add-a-search-bar-to-wordpress-menu-step-by-step/. The header search of the theme just an additional feature but it’s not a requirement, users could always add their own search features.
Once again. the ones you referred to those other threads are different, they are search form within the search page and have no relation to header search.
Thanks!