The suggestion that if one pushes away something by throwing something else builds on a purely classical intuition and wouldn't require quantum mechanics to explain if this was all we observed. The uncertainty in quantum mechanics is fundamental (to quantum mechanics) and emerges through a different, as yet unknown, mechanism.
> emerges through a different, as yet unknown, mechanism.
In 3Blue1Bron's explanation[1], he shows how the intrinsic uncertainty is an inherent trade-off of trying to measure both position and frequency. A short wave packet only a few wavelengths long correlates with a narrow (precise) range of positions, but also correlates well with a very wide range of frequencies due. A Heisenberg-like uncertainty exists any time you are working with weave packets with length near the wavelength. 3Blue1Brown gives a very good example using Doppler radar.
When I suggest the mechanism is unknown, I mean that Heisenberg uncertainty is a postulate of quantum mechanics. In other words the fundamental reason that quantum mechanics should appeal to wave mechanics isn't really established - we don't really know yet the fundamental objects and interactions that lead to quantum mechanics (despite much effort).
Moreover, similar calculations can be done with other measurements that don't conmute. One that is very important is the spin of a particle in the x, y, and z axis.
Another is the polarization of a photon in directions that are at 45°. For example, most of (all?) the experiments of the EPR paradox are done with polarization instead of position-momentum, because polarization is much easier to measure. https://en.wikipedia.org/wiki/EPR_paradox