The short answer is : it is a fundamental property of nature.
The very short answer is "quantum"
The long answer:
From the beginning of the 20th century, slowly but certainly Nature revealed to us that when we go the very small dimensions its form is quantum. It started in the middle of the nineteenth century , with the table of elements which showed regularities that could not be explained except by an atomic model with equal electrons to the charge of the nucleus.
There were efforts to understand why the electrons which were part of the atoms did not spiral down into the nucleus and disappear, with the Bohr model . This introduced the idea of "quantum". The energy the electrons were allowed to have in the possible orbits around the nucleus was postulated to be quantized. In a similar way that the vibrations on a string have specific frequencies allowed with wavelengths which are multiples of the length of the string, the electrons about the atoms could have only specific energies.
Then a plethora of experimental results led theorists to postulate quantum mechanics from a few "axioms" . Starting with the Schrodinger equation formal theoretical quantum mechanics took off and we never looked back because it fits perfectly all known experimental data in the microcosm, and not only.
The uncertainty principle is a lynch pin in the mathematical formulation of quantum mechanics.
A premiss is that all predictions of the QM theory are given as probability distributions, i.e. no observable can be predicted except as a probable value.
In quantum mechanics to every physical observable there corresponds an operator which acts on the state functions under study. Operators often are represented by differential forms and the algebra of operators holds. In quantum mechanics two operators can be commuting, that is they can be like real numbers a*b-b*a=0, or not, the value can be different than 0. This means that one is working in a larger set than the real numbers, complex numbers are needed.
The Heisenberg uncertainty principle for position and momentum as it appears in the fundamental postulates of quantum mechanics is a commutation relationship between conjugate variables, x and p, represented by their corresponding operators:
This relationship is very fundamental in the theory of Quantum Mechanics which describes very successfully matter as we have studied up to now, mathematically. If the HUP were falsified it would falsify the foundations of QM.
Now on the subject of the electron and the nucleus. The quantum mechanical solutions that describe the orbitals of the hydrogen atom, for example, have non zero probabilities for the electron to find itself in the center of the nucleus, when the angular momentum is zero. So it is not clear to me how Feynman could have used that hand waving argument you are describing in your question. After all we do have electron capture nuclear reactions. He is probably basing the argument on the very small volume the nucleus occupies with respect to the atomic orbitals which will give a very small probability of capture.