Right. It is a direct result of the basic principles of quantum mechanics that the narrower the probability distribution of one obervable is, the broader that of it’s conjugate will be.
However the time-energy inequality is not on the same footing as the position-momentum uncertainty principle since there is no Hermitian operator corresponding to time. Yes, time is a dynamical quantity in that it varies with time, but in a trivial self-referential way and is really just a parameter on which other quantities depend. The actual meaning of ΔE and Δt in the time-energy inequality are respectively the spread in the energy distribution and the amount of time it takes for the wavefunction to changed appreciably.
…therefore contradicting the tenets of quantum mechanics. Fine. But I would apply the term “on a much deeper level” instead to the relation between the uncertainty principle and the principle of complementarity.
When due to the basic principles of quantum mechanics the use of one classical concept excludes the use of another, they are said to be complementary. The principle of complementarity says that the experimental arrangements that measure complementary properties are mutually exclusive and are both needed to demonstrate all of the physics of quantum mechanical systems.
For example, consider wave-particle duality as applied to an electron which is the first form in which one usually encounters the concept of complementarity. Wave-particle duality is often erroneously described as meaning that the electron is simultaneously wave and particle. But this is impossible since particle and wavelike characteristics are strictly incompatible. What saves us is the uncertainty principle which says that there are no experiments one can perform in which the position of the electron, this being the particle aspect, and the momentum of the electron, this being the wave aspect, can be simultaneously measured to arbitrarly high precision.
Thus the deeper meaning of the uncertainty principle is that it is the condition that ensures the logical consistency of quantum mechanics.
A unitary quantum theory is one in which probability is conserved. Though they're pathological, one can imagine nonunitary theories in which the uncertainty principle formally still holds.