I’m not really sure who “owns” the properties of the ex-Ecclesia Dei societies. I thought most were owned by the dioceses and not the societies as a condition to “operate” in those specific dioceses. My every experience with the FSSP has been in diocese-owned churches/chapels.
Without enough details for absolutes, I do know the little paragraph of history of one FSSP parish church describes the people building the church. Attributions are made to no one else, including the diocese, but to disparate “contributors”, of which the diocese may, of course, be one among them.
Presently, “the people” are now looking for new land to build a larger church, and the collections for the project have already been years in the making.