hi @Mrstep
I have reviewed your requirement, you just need to edit css script and the issue will be resolved. You can follow my instructions!
Step 1: Go to Admin → Online store → Theme > Edit code:
:Step 2: Search for the file base.css
put this code in the end
.header__search details {
display: none !important;
}
Step 3: Save and reload home page.
=>> The result:
We hope my suggestions will solved your issue.