Why does brining work with dry heat methods and not wet heat methods?
Salt dissolves in water. Brining is simply the process of soaking something in a saline solution so that either it absorbs saltwater, or the salinity of the pre-existing water approaches equilibrium with that of the brine. The end state is just a lot of water, some on/inside the meat and some outside, all with approximately the same salinity.

If you roast the result, you're evaporating most of the water but keeping the salt. If you boil it, then you're just reversing the brining process you followed in the first place, by diluting whatever salt (not to mention any other minerals/flavour) it has.

The salt in a brined chicken is not reacting with the meat, it's just dissolved in the water (which is in/on the meat). Give it a good enough "rinse" and the salt water will be gone.