Unless the costs exceed their current operational costs. Don't forget the time spent redeveloping on the new API, which includes validating everything is there, testing and cleaning up and removing the old (working) code.
This isn't a great analogy here - getting the data delivered via API is simply more useful than having to re-assemble that data out of fragments parsed off of different web calls.
Could I suggest:
"Why buy the cheese when you get the milk for free?"