I suppose the obvious answer is to require them to pay — as you say, it becomes a political problem for politicians to write law mandating it and the providers to lobby for it not to be forced.
The closest analogy I can think of is the Data Communications Company that intermediates smart meter data in the UK between the energy suppliers and end users. As a monopoly, their prices are regulated (well or not is a political problem…) and all the providers have to use them.
If the data exchange was the source of truth, the providers would presumably save some money not having to store their own copies (or at least, could store more summarised versions) with all the requirements that HIPAA/GDPR entail.
However, there’s money to be had for a huge aggregated database of everyone’s health records, stored in consistent format for easy consumption…maybe we should be careful what we wish for.