Lock hospitals had nothing to do with disease of any kind. They were institutions for sexually deviant women (categorized as "prostitutes" though the management wasn't in the business of examining how they came to be so labelled). Women were locked up in them for moral reasons, not medical ones. They survived until very recent times in Ireland as the "Magdalene" institutions, though the "lock hospital" label fell out of use in the middle of the 19th century. They don't have anything to do with this family of songs as far as I can see.

The American "Streets of Laredo" tune is clearly that of the Irish "Bard of Armagh"

No it isn't. Both of them are "The Banks of the Devon", which by the early 19th century was by far the best known name for the tune everywhere in the Anglophone world.