They don't get 'disposed of'. Whenever there are large excavations it's a planning requirement to turn the site over to archaeologists first, who can find anything that's to be found before it's dug out. E.g., there was a lot of archaeology done with the Crossrail project: https://archaeology.crossrail.co.uk/exhibits/roman-moorgate-...
Yes, but that's a large excavation for public purposes, a private project is probably smaller - so does the same rules apply? Can those rules be avoided when done privately?
I wonder how often an inconvenient archaeological site gets quietly disposed of.