This is unfortunately caused by the pop-science presentations of the uncertainty principle. To answer this fully, two different pieces have to be teased apart, and precision in language becomes important. There is a general confusion of two different phenomena. The uncertainty principle and the observer effect.

Observer Effect A phenomenon where measuring a system requires interacting with that system and changes the observables of that system.

The uncertainty principle Given an experiment where we can set the initial position of a particle to be at (0x, 0y, 0z, 0t) with a momentum p and we later measure the position and momentum of that particle a time t later, if we repeat that experiment for many particles, we will get a range of different final positions and momenta. The uncertainty principle holds that the product of the standard deviations of the distributions of final position and momenta, given a definite starting position and momentum, cannot be less than some limit. If one knows a particle's position and momentum now, one cannot predict both what its future position and momentum will be.

These two ideas (the uncertainty principle and the observer effect) are completely unrelated. Just because when a system is measured, it has to be interacted with does not mean that interaction is unpredictable. We can, in principle, fully account for all interactions and know how the measurement will affect the system and back out information about the system pre-interaction.

The uncertainty principle is orthogonal to measurement, and is related to what is possibly knowable about a system given full, perfect information. We can know perfectly where a particle is now, and what its momentum is now, and that will not allow us to accurately predict what its position and momentum is in the future.

The uncertainty principle is epistomological, it is a principle about what is fundamentally knowable. We can know everything there is to know about a system, we can fully specify its wavefunction and know how it will evolve forever and still not be able to predict the values of all observables about the system in the future. Knowledge of the future position of a particle is fundamentally incompatible with knowledge of its future momentum. Joint knowledge of future position and momentum are guaranteed to always be fuzzy with some spread in either or both observables.

There are often things reported about measurements "beyond the Heisenberg limit" which are breathlessly reported on, saying that they "broke the uncertainty limits" or whatever. Those are related to incorrect statements about the uncertainty principle and that it applies to any one measurement at all. To be clear, the uncertainty principle does not restrict what one can measure about a particle. You can perfectly measure a particle's position and momentum at any given time. The thing that the uncertainty principle forbids is using that information to know, with certainty, the future position and momentum of the particle.

> But as far as I'm concerned that is not a natural law but a restriction caused by our own inability to observe those data points without influencing the system ourselves.

To be very clear, it is a natural law that is not premised on experimental failures to observe things nor is it caused by our inability to observe without influencing. In mathematical models where all information about a system is known precisely, and all measurements are perfect, the results are the same. Future position and momentum cannot be known with joint precision below a certain limit.