As the name suggests, an entanglement breaking channel $\Phi$ is such that $(Id \otimes \Phi)[\rho]$ is always separable, even when $\rho$ is entangled. Won't such channels be useless, as they destroy entanglement? Does it make sense to say a channel that is more "entanglement breaking" is more useless?
Even if such a channel is useless, it's still worthwhile to define them for things like quantum communication, to see whether or not you can use your channel to transmit quantum information. So the channels are "useful" in the sense of recognizing what type of channel you have (mathematical formalism should always make it easier to recognize when you do or do not have entanglement breaking). The usefulness of the concept is that you can find when your channels are not entanglement breaking and thus use them to transmit quantum information.
It does not seem to make sense to me to define a channel as more or less entanglement breaking, because entanglement breaking is a binary statement. You could talk about the reduction in entanglement a generic state undergoes because of a certain channel, but then you'll have to find some notion of a "generic state" because not all states will have their entanglement behave in the same way for arbitrary channels.