Manishearth's answer is correct, and this is just a minor extension of it. Manishearth correctly points out that the problem is your statement:
There is a definine velocity and momentum, we just don't know it.
Your statement is the hidden variables idea, and courtesy of Bell's theorem we currently believe that hidden variables are impossible.
Take the example of a hydrogen atom, and ask what the position of the electron is. The problem is that properties like position are properties of particles. It doesn't make sense to ask what the position is unless there is a particle at that position. But the electron is not a particle. The question of what an electron really is may entertain philosophers, but for our purposes it's an excitation in a quantum field and as such doesn't have a position. If you interact with the electron, e.g. by firing another particle at it, you will find that the interaction between the particle and electron happens at a well defined position. We tend to think of that as the position of the electron, but really it isn't: it's the position of the interaction.
The uncertainty principle applies because it's not possible for an interaction, like our example of a colliding particle, to simultaneous measure the position and momentum exactly. So you're sort of correct when you say it's an observational limit, but it's a fundamental one.
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 the "quantum" of black body radiation to atomic orbits: 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. Transitions would release a quantum of electromagnetic energy, a photon. This allowed to explain the atomic spectra as transition between orbits.
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 premise 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 ab-ba=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:
$$[x,p_x]=i\hbar$$
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.
Best Answer
Yes it does. There is a common misunderstanding of the uncertainty principle as our own lack of knowledge. When you read poor descriptions like "it is impossible to determine the momentum and position at the same time", you may interpret "impossible" as a limitation of our knowledge or tools. It is not. The uncertainty principle means that the electron DOES NOT HAVE the exact position and exact momentum at the same time. No matter how great the tools are, you cannot measure what is not there to measure.
Why? Simple. Particles interact with each other as particles, but between interactions they fly as waves. For example, if you send an electron through a screen with two slits, it will pass through both at the same time. Look up the double slit experiment for more information.
This is called particle/wave dualism that describes the nature of our reality. Particles are not microscopic "balls". Particles are waves that only interact with each other like "balls" when they meet. Quantum mechanics is also is known as wave mechanics, as it describes particles as wave functions with quantum properties.