I think it’s kinda more that (in 3D with time moving at a constant rate) space “falls” into the gravity well, and matter goes with it. In 4D this looks like spacetime being bent.
For understanding a handful of highly symmetrical systems, it might help a student understand some intuitions about what Killing vector fields and congruences (notably those made by choosing the velocity vector field of a set of geodesics) are, and tends to lead into an investigation of what the shift vector in a 3+1 decomposition represents.
For calculating things like the spherical orbits around or the photon surface of a real black hole like our galaxy's central Sgr A*, the river model seems outright unhelpful. For example, how does a river model help to understand https://duetosymmetry.com/tool/kerr-circular-photon-orbits/ ?
> time moving at a constant rate
This is another way of saying slicing of a Lorentzian (4d) spacetime into non-overlapping spaces organized along an arbitrarily chosen future-directed non-spacelike worldline. That is, this is a 3+1 slicing. We can slice along your worldline, or on that of a neutral hydrogen atom floating in intergalactic space, or on that of a high-energy cosmic ray, or on that of a CMB photon. It's arbitrary, and each can give markedly different spatial slices through the same spacetime (in particular particle counts on slices will differ where the choices of index axes are anywhere accelerated with respect to one another).
When we decompose in this way, and take an <https://en.wikipedia.org/wiki/ADM_formalism> approach, we will tend to think of the shift vector as how we associate a point one one slice (everywhere in space at a coordinate instant in the spacetime) with its successor slice (everywhere in space at the next coordinate instant int he spacetime), which is helpful when spacetimes expand or contract in one or more spatial directions along the arbitrarily chosen time axis.
Braeck & Gron 2012 have a good bit of pedagogy about the river analogy and a fine set of references <https://arxiv.org/abs/1204.0419> and of course point to Hamilton & Lisle 2008, as originators of the analogy <https://arxiv.org/abs/gr-qc/0411060>.