To me, "undue grace" and "generosity" imply some actor who imparts such grace or generosity. But that's not required for privilege. Particularly when people talk about enjoying a privilege in the social-justice way (which I believe to the kind of criticism you're talking about), they're often using it to mean a lack of discrimination as opposed to granting an elevated status. For example, if someone has the expectation that they can walk through a store without getting harassed, we may say they enjoy a privilege, even though most people would agree that should be the norm - because there are people who do expect to get harassed. The reason I think it's important to remove an actor who grants privilege is that its causes are often structural and implicit, as opposed to intentional and explicit. And it's less a criticism than it is a way of pointing out that some people's default expectations for how the world works and will react to them is quite different from others.

Put another way, privilege is often used when some people tend to experience a fairer world than others. But expecting a fair world is how it should be! That's not "undue grace", as everyone should experience a fair world.

(Just so we're clear: this is a full-on semantic discussion, and I'm okay with that.)