the only way to measure an elementary particle is to make it interact with another elementary particle: it is therefore incorrect to say that an elementary particle doesn't have a well defined momentum/position before we make our measurement.
No it is not incorrect to say that "an elementary particle doesn't have a well defined momentum/position" , either describing before or after any measurement, because quantum mechanics is about probabilities. The electron around the nucleus is in an orbital, not an orbit. A probability locus.
We have done this single particle scattering experiment without a microscope over and over again in elementary particle interactions by scattering particles on each other and detecting their track in our chambers.We measure the crossections for the scattering, i.e. how probable it was. We have found that particles do not behave like billiard balls, which is the ultimate expectation of classical scatterings with no uncertainty. In classical mechanics we can go back and find the (x,y,z) to any accuracy also for an equally accurate momentum , not so in the microcosm of particles scattering on each other.
All this microscope business is beside the point, since the Heisenberg Uncertainty has been incorporated in the Quantum Mechanics theory: all observables correspond to operators and for some observables the commutators do not commute, these are the canonical commutation relations. .The predictions of the theory of Quantum Mechanics, which are based on the canonical commutations relations, have been fully validated and are being continuously validated. Quantum mechanics is the underlying framework of nature from which classical mechanics emerges. This Heisenberg microscope is an unnecessary excursion to the time when experimental verification was scarce .
We cannot access this data (momentum/position) without changing it, therefore it is correct to say that our ignorance about this data is not an intrinsic property of the Universe (but, rather, an important limit of how we can measure it).
This is incorrect. In classical mechanics we expect our theoretical formulas to work to any accuracy and thus any uncertainty is attributed to measurement errors. In quantum mechanics our formulas do not predict exact space and time and energy/momentum variables. They predict a probability for finding the particle with the specific value of the variables. It is an intrinsic , part of the theory , property, it has been fully validated and tells us that the underlying framework is probabilistic.
The Heisenberg Uncertainty principle does not need thought experiments since all experiments we do in the microcosm do not deviate from it.