Here an interesting case study concerns the case $M=$ Leibniz. We have undertaken some detailed studies of primary documents recently, resulting in publications in the British Journal for the History of Mathematics and elsewhere, and I can report on a few conclusions that are particularly relevant to this question.
$\Phi_{\text{Leibniz}}$ assumes the validity of the part-whole principle as applied to infinite domains. This implies that what Leibniz refers to as "infinite wholes" is a contradictory concept and therefore useless in mathematics, according to Leibniz.
In view of the above, the Peano Arithmetic would be a better starting point for a Leibnizian $\mathfrak F$ than ZF.
Working within ordinary numbers, Leibniz assumes the existence of a primitive predicate he calls "assignable". Leibniz uses the term infinitum terminatum for an inassignable number bigger than each assignable number. The term sounds somewhat paradoxical to a modern ear in English translation: "bounded infinity" (this does not of course refer to an ordinal or cardinal but rather a member of a ring such as $\mathbb Z$ which can be formalized in PA). The inverse of such a number gives an infinitesimal.
Karel Hrbacek and I have developed an effective foundation for real analysis with infinitesimals in a foundational system we called SPOT (an acronym for its axioms). This is effective in the sense of being conservative over ZF (without choice). Since SPOT incorporates ZF, it is not fully faithful to a Leibnizian foundation (see items 1 and 2).
A better foundation for $\Phi_{\text{Leibniz}}$ would perhaps be some jazzed-up version of Peano Arithmetic incorporating the assignable primitive.