The observation of annihilation relies on dark matter being constituted of unknown and maybe supersymmetric neutrinos, right?

The exact nature of dark matter (DM) does not enter into the game here. Predictions from DM-originating neutrinos from the sun only relies on the fact that dark matter can annihilate into Standard Model particles, and that dark matter can interact with protons and neutrons. These properties are there for any DM model.

Now, let's assume that one would observe nothing (which is the case at the moment). We can thus invert the problem and constrain the various dark matter models that are on the market by confronting predictions to null results.

Sneutrinos candidates are one option among others, but you need some non-minimal supersymmetric model (in the minimal supersymmetric extension of the Standard Model, sneutrinos DM are ruled out by data). Nothing too crazy actually, like left-right supersymmetry or supersymmetry with a fifth force.

But could Dark Matter also be due to some yet unknown extension of the Standard Model or is there some fundamental argument saying that there can't be anything beyond the Standard Model?

The Standard Model does not explain dark matter. Therefore, you need to go beyond the Standard Model and complement it by at least a DM particle, and some mechanism connecting the DM particle to the Standard Model. That is the minimal way. Of course, most models are by far not minimal :p

Don't hesitate to come back to me if you have any further question.

PS: I still don't know your institute :p