wp_authenticate but not logged in

wp_authenticate() just checks user data, but not actually authenticating – docs.

You can use wp_signon, which uses and wp_authenticate: source, like this:

    $credentials = [
        'user_login' => $name,
        'user_password' => $password,
        'rememberme' => true,
    ];

    $signon = wp_signon($credentials, true); // true - use HTTP only cookie

    if(is_wp_error($signon)){
        return false;
     }
     // The user is logged in redirect, return true, etc.