You substitute an invented term "old songs and ballads" for the correct one "folk song". That adds nothing and loses much since it loses the correlation between folk song and folk music on the one hand and folk arts and folklore generically on the other. Schmaltzy Matilda (I like the coinage) is a fine song in its own way and there's nothing inherently evil in singing it - and if your own arguments are right then it is a traditional song. Reductio ad absurdum.