True, but of course also the functioning of the accelerometer needs to use the fundamental laws. At the end it's a question of consistency and the limits of accuracy you choose to achieve or are able to achieve. You cannot get out of this dilemma that you use the very laws to construct measurement devices you want to test using these devices. At the end it's a question of consistency whether a model like Newtonian mechanics is a valid model to describe the observed phenomena or not, and you cannot even start to measure anything without an assumption of such a model.
Indeed, without much thinking usually you start to use the Earth as a refrence body. Say you want to measure the laws of free fall (because it's clear that there's always the gravitational interaction of anybody with the Earth; the unavoidable gravitational interaction between the bodies can usually be neglected, but that's another issue, but it's also an issue of accuracy). So you take the Earth as the reference body and take a ruler to measure vertical distances ("vertical" defined as the direction of the gravitational force/acceleration, which we can assume as homogeneous for not too large spatial regions we perform our measurements in; from Newton's Law of gravity we know the relevant length scale here is of the order or the radius of the Earth; so in usual lecture-hall experiments it's safe to assume that the gravitational acceleration ##\vec{g}## is just a constant). Now you realize that anybody falls in a straight line along this ruler. You can measure distances with this ruler (say from the starting point of the falling body to the floor of your lab), because you assume (!) the validity of Euclidean geometry for the physical space.
Now you also need a clock. One way is of course to use a mathematical pendulum (or to be more accurate Huygens's isochronous pendulum with a body moving along a cylcloid to avoid a dependence on the amplitude). Newton's Laws (which are again assumed here!) lead to a measure of time due to the formula ##\omega=\sqrt{g/l}##, for which you only need to be able to measure the length ##l## of the pendulum. The value of ##g## is not so important, because you can take it as a constant around you lab, and you can simply define a measure of absolute time by taking the frequency of a certain standard length to define your unit of time. Of course you can now check the law for different pendulums by measuring the frequency as a function of ##l## using such a standard clock to compare the frequencies of these various pendulums.
Now we have established a measure of time. Now of course you can measure the trajectory of bodies as a function of the so quantified time ##t##. Of course, you'll then find with more or less accuracy, the assumption ##\vec{g}=\text{const}## confirmed when measuring freely falling bodies has a function of the height ##x=h-g t^2/2## (if you always start with 0 initial velocity).
As a next step you can also check the law for a body moving not only along the "vertical". If you neglect air resistance of course you get the law that velocity components of moving bodies perpendicular to the direction of ##\vec{g}## stay constant. This pretty much ensures you that taking the Earth as a reference body is a good approximation of an inertial frame, where of course you have to take into account the gravitational interaction between the observed bodies and the Earth.
There are of course a lot of assumptions and some math going into it like solving the equations for free(ly falling) bodies or a mathematical pendulum. Then you can make all kinds of measurements and check whether all this assumptions stay consistent with each other and in accordance with the predictions using Newtonian mechanics.
As stressed before, it's also a question at which accuracy you measure. Of course we all know that the Earth is not really a reference point to define an inertial frame. The Foucault pendulum experiment demonstrates that it is a rotating frame as the correct prediction of the precession of the pendulum's plane of oscillations assuming the corresponding corrections due to the inertial forces (here the Coriolis force is sufficient) expected in a uniformly rotating reference frame (i.e., the Earth rotation around its axis).
Also the assumption that the bodies are not mutually interacting is of course only an approximation, and indeed Cavendish managed to demonstrate Newton's universal gravitational law and to measure with some accuracy Newton's universal gratvitational constant with his experiment torsion balance using the mutual gravitational interactions of bodies (at an accuracy within about 1% compared to the modern value).
Of course, as an empirical science physics cannot be axiomatized. It's always an interplay between theoretical thoughts and the construction of measurement devices to make quantitative observations. The "natural laws" are always subject to tests and consistency checks with the underlying theoretical assumptions.
The history of physics shows that indeed adjustments to the very fundamental laws are happening, though in an amazingly slow rate. Newton's mechanics was consistent for around 200-300 years until the spacetime model finally turned out to be only an approximation, when in 1905 Einstein discovered special relativity and thus introduced the more comprehensive Minkowski-space spacetime model, which was consistent with a larger realm of phenomena, i.e., not it was consistent with both mechanics and electrodynamics. Just 10 years later the spacetime model had to be adjusted again, because it turned out that gravity could be most easily described by reinterpreting it as a curved spacetime manifold and at once unified the phenomenon of inertia and gravitational forces (though only in a local sense of course). Though General Relativity (GR) up to now stands to be highly accurate, it may well be that one day one needs even more accurate laws, and that's why GR is also tested with ever more accurate measurements as the advance of technology allows.