Just for fun, and to highlight what is dangerous with the sort of implicit assumption you are making with statements like "You can't prove it's older than 1852" based on the date of the earliest written source, here's a little non-folk problem to ponder.

The Wall Bedstraw (Galium parisiense) is a small plant of the Bedstraw family. It was first recorded in Tottenham by John Sherard in 1690 and, though nationally scarce, can still be seen in Tottenham only a few yards from where Sherard spotted it.

Note what this tells us about the Wall Bedstraw ... about the only thing it proves ... is that it was there before Sherard spotted it. Similarly, knowing the publication date of the earliest manuscript source of a song can only tell you that it's older than that. Of course, it doesn't tell you how much older, but - as with the wall bedstraw - you then have to exercise your judgement about it, or (even better) use some other different evidence.

;-)