How do you disable the verification process of user email changes?

As mentioned in the ticket #16470 this confirmation feature:

prevents accidental or erroneous email address changes from
potentially locking users out of their account.

so it’s good to keep that in mind before removing this feature.

Here’s a little proof of concept for (single site) test installs.

We still want to run send_confirmation_on_profile_email() because it contains various email validations and error reportings, but we want to stop it from sending a confirmation email. Instead we set the generated _new_email meta value from that function as the new user email during the current user profile saving.

Here we unhook send_confirmation_on_profile_email() and rehook it within our wpse409429_update_profile_email() wrapper with our modifications:

add_action ( 'admin_init', function() { 
    remove_action( 
        'personal_options_update', 
        'send_confirmation_on_profile_email'
    );
    add_action( 
        'personal_options_update', 
        'wpse409429_update_profile_email'
    );
} );

where our wrapper is:

function wpse409429_update_profile_email( $user_id ) {

    // Stop confirmation email when running 
    // send_confirmation_on_profile_email().
    add_filter( 'pre_wp_mail', '__return_false' );
    send_confirmation_on_profile_email();
    remove_filter( 'pre_wp_mail', '__return_false' );

    // Get new email that is already validated and saved
    // as _new_email within send_confirmation_on_profile_email().
    $new_email = get_user_meta( $user_id, '_new_email', true );

    if ( ! empty( $new_email['newemail'] ) && is_email( $new_email['newemail'] ) ) {
        // Core uses this output! escaping during saving, so we do the same!
        $newemail = esc_html( trim( $new_email['newemail'] ) );
    
        wp_update_user( array( 
            'ID'         => $user_id,
            'user_email' => $newemail
        ) );

        // Core modifies the _POST value for email, 
        // with the old email value, so we need to 
        // override that with the new one.
        $_POST['email'] = $newemail;

        // Delete post meta to avoid confirmation messages.
        delete_user_meta( $user_id, '_new_email' );
    }
}

Further one can add extra error reportings if meta value is corrupted and hide the text under the email field:

If you change this, an email will be sent at your new address to
confirm it. The new address will not become active until confirmed.

with CSS.