Searching without standard s input field

I am trying to make a search form without the use of plugins (so please dont point me towards one) that will function without the s input.

I am able to get this to work when there is s input…but I just dont want that field there, and I don’t want to just hide it via CSS…I want it gone entirely.

When I remove it, the search function no longer works at all and just refreshes the page I’m on. What’s even stranger is that if I keep the search text field empty and leave it on my form, I get routed to the index.php template file with the results I want…but I want to use search.php. Search.php is loaded if I actually put something into the search field (not leaving it blank).

Could anyone explain to me how this is working and how to remove the standard search input the correct way?