when I added recaptcha 3 it appeared on all pages

 reCAPTCHA’s script file is loaded on every page, can I stop it?

If there’s no way to stop this it’s not worth using the plugin. Right now I have it deactiated. UI / UX for users matters. I have recaptcha on login page for a different plugin and that doesn’t cause the entire site to permanently have the badge.

All that would be enough would be a built-in option to move it to another side and/or having the ability to choose if one wants the badge or just the text with links below the form.

Ok so I’ve managed to move the badge in a relatively nice way, based on the script handle name:

add_filter( 'script_loader_src', function ( $src, $handle ) {
    if ($handle === 'google-recaptcha') {
        return add_query_arg( 'badge', 'bottomleft', $src );
    }

    return $src;
}, 10, 2 );

In reality it adds “&badge=bottomleft” to the https://www.google.com/recaptcha/api.js safely – in case there are more params (and there are more params).

Thanks for the code but I’ll probably just skip this plugin. Just not worth it for UX to have a permanent reCAPTCHA badge on all pages

Google tells you how to hide it and what you have to do if you do that:
https://developers.google.com/recaptcha/docs/faq#id-like-to-hide-the-recaptcha-badge.-what-is-allowed