In a "modern way of teaching Newtonian mechanics", there are several notions to define before stating Newton's laws, which appear in the laws.
(i) Position and time in reference frames. I'll consider that it is possible to measure it so that acceleration in a given reference frame is computable.
(ii) Mass should be defined. How can we define such a notion? How can we measure it?
(iii) Force is to be defined too and, of course, a way to measure it.
My understanding of Newtonian's law is the following (I will not talk about Newton's third law):
(1) An inertial reference of frame is, by definition, a frame in which if there are no net forces, that is, if the sum of all forces acting on an object is equal to zero, then $a=0$.
(2) In such inertial frames, $ma=F$.
(3) The action-reaction law.
These statements are somewhat "circular", this is no news as many other questions on this SE can attest this fact.
It seems that the second Law (2) is a definition of both force and mass: you measure a and you observe that there is a proportionality constant $m$ depending on the object such that $m_1a_1=m_2a_2$ when the two objects are in the same conditions. With this notion of mass, you identify $F$ as the thing equal to $ma$.
But in order to do so, you need to work in inertial frame of references ... which requires the notion of force to be defined! And the force is also by definition ma in intertial frames ... this seems circular. How do we make it uncircular?
Is there a rigorous construction of mass, force, and Newton's first/second law?
Let me give a troubling example, may somebody explain to me why I am wrong: Foucault's experiment is considered as a proof that the frame of reference given by the earth in which we are sitting in, is not an inertial frame. To do so, one cooks up an inventory of forces, let us say that all these forces add up to $F$. Then one observes the acceleration of the pendulum $a$, and observe that $ma\neq F$. To restore the equality one has to add a "fictitious" force, $F_f$, so that $ma=F+F_f$. My objection is then: why do we say that this frame of reference is not inertial, rather than admitting that we forget a force in our inventory?
Just to be clear, I am not a first-year undergraduate student who does not understand anything to Newtonian physics. I am pretty convinced that the earth is not an inertial frame of reference and pretty aware that in the appropriate frame of reference in which Earth is rotating, the "fictitious force" is just incorporated in the acceleration. But ... Newton's law (2) states that "acceleration is force".
If you think that this question needs modification or clarification, do not hesitate to tell me. I should mention that I did not find any other questions on the Physics Stack Exchange properly addressing my question.