I think you would need the search form on that page yes. On the results page, I normally hide any general theme search functions, and have just our form (even if the search is in a nav bar, I would hide that on the search results page).
If the search results page (that should have our form on) is to be the one you want to use even for the general theme search, you can use a WordPress rewrite to make that general them search send a text search to our form.
In the child theme functions.php file you would have a function like this:
function sandf_change_search_url_rewrite() {
if ( is_search() && ! empty( $_GET['s'] ) ) {
wp_redirect( home_url( "mypageslugname/?_sf_s=" ) . urlencode( get_query_var( 's' ) ) );
exit();
}
}
add_action( 'template_redirect', 'sandf_change_search_url_rewrite' );
Where you change mypageslugname
to that for your search results page.