Moreover, in "Does provable absence of barren plateaus imply classical simulability?" it's argued that while QCBMs can avoid barren plateaus, they can also be efficiently simulated via TNS, if I'm not mistaken.

The heavy lifting is definitely done using classical machine learning. When it comes to the QCBM, consider that calculations are done on a 16-qubit system. If this quantum prior is actually useful remains to be seen (especially when going to larger systems).