> slower and slower

And grow dimmer and redder and blurrier.

There are several assumptions here, mostly having to do with the exterior Schwarzschild solution to the Einstein Field Equations of General Relativity serving as the source of most non-mathematical descriptions of black holes. The crucial assumptions: the black hole is eternal; it is maximally symmetrical (eternal means it's time-symmetrical, and the horizon being perfectly spherical gives spatial symmetries - let's also add in an absence of angular momentum and any non-gravitational charge); and it is the only substantial mass in its universe. "The perspective of their family" needs to be accelerated with respect to the central mass, for example by hovering directly over a point above the (non-rotating, spherically symmetrical, neither growing nor shrinking) horizon. The dying people also have to be in radial free-fall, so "throw" means just that[1], rather than a rocket-powered plunge or an eccentric hyperbolic trajectory.

Breaking these conditions may provide the family of the infaller a very different view (for example, you could rocket the family in question into the same, or another, black hole such that they hit a central mass before light image of their infalling relative does; alternatively you could in principle, by throwing in other matter, make the black hole's mass and/or angular momentum grow quickly according to the watching family's wristwatches).

> like it's not there

The "no drama" conjecture for a freely-falling infaller depends on the global spacetime, and in general requires an arbitrarily large value for the central mass, so that the curvature scalars at the horizon are arbitrarily small. One can construct realistic black hole entries which are promptly extremely noticeable for the unfortunate infaller (or indeed for the black hole; infalling large masses will effectively raise bumps on the horizon that require on the order of a horizon-diameter light-crossing time to settle down by way of gravitational radiation, as is looked for at LIGO and Virgo).

Inconveniently for understandings based on the Schwarzschild solution, lots of astrophysical black hole candidates appear to have enormous angular momentum. Additionally, they certainly aren't alone in our universe. Probably all of them are also still increasing in mass and angular momentum as things including distant starlight, cosmic microwave background and cosmic neutrino background radiation (and in the cases of astrophysical black holes in galaxy clusters dust, gas, stars, other black holes and so on) collide with them. This breaks several of the symmetries that lead to predictions of observations of an arrangement like M<->h<->i<------->a where M is the central mass, h is the horizon, i is the infaller freely-falling towards M, and a is the accelerated observer not moving towards M (it can however move away from M, relativistically if it wants[2]), all of which have Schwarzschild coordinates that differ only in "r".

Finally, it's hard to find an astrophysical black hole without any sort of accretion structure around it, because isolated black holes are so very nearly black. It'll be harder to find an accretion structure which wouldn't obscure the view of the dying people in various ways as they approach the horizon, not to mention that our distant observer would almost certainly see a moving image of them being killed by bombardment with high energy electrons and X-Rays [ https://www.youtube.com/watch?v=-OtUVDRL_wM ] well before their well-cooked corpses started to resemble a photograph.

(It's also worth observing that the mass of an accretion structure is going to be much much larger than that of a spaceship full of dying people, as will the flux of mass falling onto plunging orbits. The large mass in the accretion structure will pull points of no return away from where they would be if the black hole were isolated from that mass; this is another aspect of raising bumps on the horizon.
The large amount of inward-plunging mass will grow the horizon beyond where it would be for the case of an isolated black hole into which one throws a ship of sick people, much more quickly than "forever", even for a distant observer.)

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[1] equip yourself with a smartphone with accelerometers, and an application like https://phyphox.org/ and throw your phone, and note what its accelerometer reports when you simply drop the phone versus when you throw the phone into the air, across a room, or into the hands of a collaborator.

[2] You might consider a central black hole in a galaxy cluster at high redshift (i.e., billions of parsecs away), whose surrounding luminous matter is already seen by us as redder and dimmer because of the metric expansion of space, versus flying in an ultra-relativistic rocket towards a nearer black hole. Less extreme interactions of translational, accelerational and gravitational redshift interactions are dealt with in GPS [ http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.... for some numbers especially at footnote 2] and other GNSS systems.