If, for outside observers, infalling matter never enters the black hole, what happens to the matter's gravity?

Although I haven’t seen any calculation of this, I think that the gravitational field of a small perturbing mass outside a black hole actually becomes centered on the hole, not on the small mass, as the mass approaches the event horizon.

I think this because something similar happens with a point charge held stationary outside a black hole. (I have seen this calculation.) As the charge is held closer and closer to the horizon, its electrostatic field becomes spherically symmetric around the hole.

My way of thinking about this is that it is part of the “no hair” aspect of black holes. If the field perturbation did not become spherically symmetric as the mass or charge approaches the horizon, then by continuity it would not be spherically symmetric when the mass or charge is inside the horizon, and we would be able to tell where the charge or mass is inside the hole. Instead, by the time it crosses the horizon (which it does, according to its own clock) it has simply added to the hole’s mass and charge.