Do physical models give a good view in the physical reality?

There are various physical models to make visible how a theory works, for example:

Falaco solitons

Water models of black holes

A helium model of the early Universe

Models how to visualize kinetic theory

The well-known example of a stretchable rubber sheet to envision the way general relativity works

And there are of course many more examples. But in how far do these models give a good picture of reality (the rubber sheet model certainly does not)? It depends on the theory, which can be right or wrong or of which this isn't clear yet (also, what once was thought as a true theory can later be considered as wrong, and vice-versa). Phlogiston was once thought to be a real substance and it was seen as a real stuff flowing into or out of a substance. Hans Radder in his treaty "The Material Realization of Science" treats phlogiston (of which the Higgs mechanism is a oresent day example) as a real stuff in the domain where it makes experimental predictions that can be realized. So phlogiston is real if the experiment can be described by using it and predict the correct outcome. Raising the new question which labels, as Radder calls the entities for describing reality, are the real ones. Phlogiston or heat? His attitude seems to be an agnostic one but there is a connection with the reality of the experiment (as there is in treaties by Hacking and Pickering).

Are these models merely an aid or is there a "real" connection, correspondence or whatever you call it, to the world? Or is the only real thing the math of the theories (as propagated by Max Tegmark). I guess we (almost) never can tell (or is it the opposite?), but answers can be given and it is certainly discussable.