Often it's a donation from an estate. I believe they discuss this in the podcast episode (though it has been years since I listened to it).

To a certain degree, it's not impossible that museums are either competent enough to recognise talented artists early, or so powerful that their buying itself is the signal that makes an artist.

Either way, it's at least theoretically possible that they could finance themselves by, essentially, savvy investment in artworks.

Many museums started in the times of yore, and became first tier through donations by powerful and rich patrons (and/or by the state).