Can I use an air to air heat exchanger to fix a radon problem?
You are much better off preventing it from entering in the first place. But once it does enter, ventilation is the only way to reduce the levels. How effective this would be depends on the exchange volume. You would be committed to running this exchanger 24/7 for ever. Replacement costs would be much higher than the typical radon fan. Anything running 24/7 will wear out eventually.

And there is still a significant energy loss running like that at decent volume. They are no where near 100% efficient.

If the floor was poured over a layer of gravel, a retrofit sub slab extraction system would likely be successful. If it was poured on compacted earth, a successful system is much less likely. You are right, just sealing cracks without any other remediation would not be very successful.

Testing for adequate sub slab permeability isn't too involved, drill a few well spaced holes through the slab and see if sucking air out of one hole draws air into the others.