The following wasn’t wrong for me:
$data = $wpdb->get_results( $wpdb->prepare( "SELECT * FROM $dbtable WHERE name = $name" ) );
It should be:
$data = $wpdb->get_results( $wpdb->prepare( "SELECT * FROM {$dbtable} WHERE name = %s", $name ) );
Then you can just die();
at the end of the function like:
if( count( $data ) >= 1 ){
die( 'true' );
} else {
die( 'false' );
}