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What is the real reason that make us reject singularities everytime we see them in a theory/model?

For example, in GR, it is predicted that black holes singularities have infinite density. This makes GR useless to predict the behaviour of the singularity.

There are some proposed alternatives to study the behaviour of singularities and all of them, assume that singularities do not have infinities, or even, "delete" singularities, replacing them with something different, for example, "tunnels" to another universe (see Black Hole Cosmology https://en.m.wikipedia.org/wiki/Black_hole_cosmology)

But why does this happen?

I've been told that when we say that singularities "break laws of physics" what we actually mean is that GR does not apply there, but another version of GR or another set of laws different from GR and our standard model would apply there. That physicists avoid singularities (or, at least "infinities") because it would really/actually break all the laws of physics and could not be described by any theory (not GR, neither a different version of GR nor a different model/set of laws of physics than our standard models...etc), and because of that, they model singularities with no "infinities" (or even they do not model/consider any singularity at all) in their works/theories.

In summary, that singularities (with "infinities") in, for example General Relativity, are avoided/ignored or thought to not exist because if they would actually have infinities, they would really/actually break literally all laws and could not be "governed"/described by any model/theory. For that reason, they are considered to be not real, having finite densities or being replaced by something different.

But is this right? What is the real cause that makes us reject singularities (with "infinities")?

When you ask most working physicists this type of question, the answer you get tends to be an oversimplified one that is partly just based on experience and conservatism. There were singularities and acausal behaviors in the classical electrodynamics of point particles, but this eventually got pretty much cleared up by QED, so the moral that people took to heart was that this was how to look at all singular or acausal behavior, as just a sign that the theory was incomplete. If you look at the work of specialists in relativity, you will find a more more complicated description. A good discussion of this kind of thing is given in Earman, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes, which Earman has made free online. Earman is both a physicist and a philosopher, and the book contains some of each.

We would like to be able to use GR to make predictions. There are some things that we don't expect the theory to be able to predict, such as what the big bang should have looked like, so the feeling is that a past spacelike singularity is OK. This gets formalized in the notion of a globally hyperbolic spacetime. A globally hyperbolic spacetime is one in which we can give initial conditions on a spacelike surface (called a Cauchy surface) and then evolve the field equations forward and backward in time. Basically we have global hyperbolicity if there are no closed, timelike curves and no naked singularities.

A naked, i.e., timelike singularity breaks global hyperbolicity because if you try to draw a Cauchy surface on a spacetime diagram, you can't, because the singularity makes a hole in the topology.

Black hole and white hole singularities don't break global hyperbolicity because they're spacelike singularities that are either in our future or in our past, never both.

If you have timelike singularities, then physics has serious problems with prediction, as shown by Earman's memorable figure and caption below:

The worry is illustrated in Fig. 3.1 where all sorts of nasty things -- TV sets showing Nixon's 'Checkers' speech, green slime, Japanese horror movie monsters, etc. -- emerge helter-skelter from the singularity.

The reason that GR can't predict what comes out of the naked singularity is simply that we can't even formulate the initial conditions in an appropriate way, because a Cauchy surface doesn't exist.

It's actually quite possible that cosmic censorship fails in realistic gravitational collapse: What is the current status of cosmic censorship? "Quite possible" doesn't mean that I or any physicists I know want to bet a six-pack on it, but that there are serious researchers who think this is a reasonable possibility, and they've suggested actual observations to check. If this turns out to be true, then generically, such scenarios lead to consequences like the release of arbitrary information and infinite energy from the singularity, which is just a polite way of saying green slime and lost socks.

People who don't like this possibility pin their hopes on the fact that there is a Planck scale where we know that GR becomes inconsistent anyway, and definitely needs to be replaced by a theory of quantum gravity. However, I don't know of any argument that convincingly suggests that in a theory of quantum gravity we necessarily recover predictive ability.