This foundation (the primitive concepts and most of the axioms based on them) is indubitable, and hence unassailable.
To see why, one would try to doubt it. However, to doubt is to entertain the possibility that an assertion is false. Already, the concept of truth, falsehood, and contingency (the possibility to be false in at least one possible world) are presupposed. With them, come the property of being true and that of being false, thus the concept of properties and and abstract objects in general.
In short, doubting presupposes:
The concept of truth and falsehood
Properties as abstract objects
Contingency and possible worlds
Another source of the primitive concepts is the doubter themself. The doubter wouldn't deny that that they are doubting, and so they must exist, in some way, in order to doubt. With this, the concept of existence and the actual world containing the doubter are obtained.
Without this Foundation, how can one conclude that the so-called circularity and begging the question are invalid (what is so "bad" about them?)? What's wrong with begging the question and how do you know, using what criterion?
And if you are someone who publishes academic papers in scholarly journals arguing for theses, publishes books, or publishes blog contents online, why should I accept any of your theses if you deny this foundation? If your sole purpose is to convince your readers of your theses (what Latour (1988) has called 'infra-reflexivity' (p. 166)) , you can simply appeal to emotion or use deception for doing so, or simply make sure that your audience is composed of certain types of individuals only. Even then, you're still believing in the possible SoAs that your texts would be read.
And if your intention was to just write a piece of text, why publish the writing in a journal at all?
In summary:
This Foundation is indubitable because doubting requires/presupposes this very Foundation.
The Agrippan trilemma and the Problem of the Criterion already presuppose a foundation.
It is similar to a massive piece of a solid substance that only exist as this single piece, and this single piece can be broken only via hitting it with the very substance which it is made of.
Without logic, one cannot prove any thesis.
Thus, one can say of this Foundation that it transcends doubts.
This is similar to Descartes's 'cogito ergo sum', the conclusion that 'I think, therefore I exist (as the object that thinks)'.
Latour, Bruno (1988). 'The politics of explanation - an alternative.' In Woolgar (1988) eds. Knowledge and Reflexivity (155-176).