It's true that object in freefall being attracted towards a gravity well does not have less mass as it approaches deeper parts in the well. However, it will have less rest mass from its potential energy, and more mass due to its kinetic energy. If the object continues on a trajectory past the gravity well and out to space beyond, it will lose that kinetic energy, and regain the potential energy. So in that sense, it's true: the overall mass doesn't change.

But if the object smacks into something, like an accretion disk around the black hole, or the atmosphere of a planet, then the kinetic energy escapes the object, and is lost to the surrounding system (through mechanisms like friction and compression, as you've mentioned). Per Einstein, this lost energy really does mean the object has lost mass.