The clue:
    The uncertainty principle assumes that both parameters are measured simultaneously.

Missed:
    Measures often have several inseverable parts.  When only one part is given, the measurement cannot, in general, be used  in calculations used to predict outcomes.
    The uncertainty principle addresses pairs of parameters and states that the product of the uncertainties in measurement of the two parameters can never be less than some number which is a universal constant of nature (1.0546E-34 joule sec).  The pairs are: 1) position & momentum; 2) angular position & angular momentum; and 3) energy & time.  We can determine the position of a pitched baseball, but if we don't simultaneously determine its momentum (mass times velocity) we cannot interact effectively with it (hit it with our bat and slam it deep into center field where the center fielder is busy tying his shoelaces).
    This need for complete information carries over from classical mechanics into quantum mechanics in the uncertainty principle.  In classical mechanics nothing limits the precision—the information content—of measurements except the skill with which we design our measuring instruments.  In quantum mechanics there is a fundamental limit to precision of measurement, to the information content of measurement.  All observations, classical or quantum mechanical,  point to limits of precision: infinite information content is only wishful thinking.
     However, quantum mechanics finds a limit imposed by the wave nature of particles, and our wishful thinking is thwarted.  Observations that lead to effective use of measurements require completeness of components, both classically and quantum mechanically.  But quantum mechanics goes one step further: it shows us a fundamental limit, a fundamental "information content."  This is simply the number of possible distinguishable states (paired values of, say, position and momentum) for the thing we are examining (perhaps an electron confined in some sort of a box).  When we know this "information content," we can calculate probabilities of all sorts of things (like chemical reactions or osmotic transfers through membranes, for example).
      And Ludwig Boltzman gets his epitaph, written on his tombstone:

     The "W" is the probability we will use in  calculations (and can be taken as that aforementioned number of states).  And the "S" is Erwin Schrödinger’s central concept in his answer to the question, "What is Life?"  It is entropy.  Living organisms are based on abilities to select from alternatives of actions so as to anticipate the outcomes of those actions.  Our world is statistical.  Life's outcomes, like outcomes in a casino, are statistical.  Entropy is a statistical concept central to predictability of outcomes of events at the molecule and particle level.  And the uncertainty principle gave us a brand new element in our understanding of entropy and a brand new way to calculate entropy.  But it's through insights at the edge of human comprehension.

And...
    The "can't simultaneously measure" error is often used to support beliefs in pseudoscientific hypotheses such as mental spoon bending and telekinesis.  The argument goes generally something like this: we can arbitrarily choose to measure position precisely, then the momentum can be anything and so doesn't really exist; it was a mental act on our part that destroyed momentum; therefore, our mental act affected the reality outside our minds.  Many errors contaminate this reasoning.  Here are a few that are slightly off the more common paths of explanation:
    Firstly, to go to the limit of perfect precision has no meaning: it would be infinite information content, an extrapolation all the way to an unattainable limit.
    Secondly,  it ignores the fact that the "W" derivable as described above is independent of any "choice of precision" of position or momentum by itself and is a function of only a complete measurement, position with momentum.  (What's important is the area in the "phase space" of position - momentum.)
    Thirdly, it ignores some rather esoteric, but beautiful, mathematics appropriate to wave (or quantum) mechanics in which position and momentum are "Fourier transforms" of each other (either is the "spectrum" of the other) and which renders the uncertainty principle simply a logical consequence of the wave nature of matter.
     Nextly, the discovery of the uncertainty principle and the derivation of the appropriate mathematics of quantum mechanics was based on centuries of meticulous observation and extensive interpretations at the edges of human comprehension, while the pseudoscientific hypotheses tend to be contrary to observation and rich in wondrous wishes...largely without the multiple mathematical ("logical," "information processing",...)  insights in and about the edges of human comprehension.
     And these are just some of the more elementary errors.  Good theoreticians can, and often do, come up with many more.

The clue:
    The world was well known to be statistical long before quantum mechanics.

Missed:
    The importance and ubiquity of the statistical nature of the world, whether quantum (modern and microscopic) or classical (recent past and macroscopic).  The nature of statistical phenomena and statistical reasoning.  (Statistics is characterized by multiple cause and effect, along with randomness and unpredictability.)

And...
    An implication is being improperly inverted when quantum mechanics being statistical is seen as suggesting that other branches of physics are not.
     Statistical principles are most often seen as nothing more than tools of  rogues and liars—which is another way of saying they simply aren't seen.  This "blindness" to the statistical is a rich source of raw material for casino-operating rogues and advertisement-writing liars.

The clue:
    Measurement would disturb the object measured whether or not the uncertainty principle is true.

Missed:
    The true "weirdness" of quantum mechanics and the uncertainty principle, which is the wave nature of all matter.  Also, the impossibility, in principle, of "absolute precision," which would mean infinite information content. [More about this]  The disturbance caused by measurement is not unique to quantum mechanics—to so see it is to improperly invert an implication.  Classical mechanics would allow us to calculate the disturbance and determine precise values existing before the measurement; quantum mechanics does not.

And...
    Measuring always involves an interaction with the object measured.  The "gentlest" measurement is probably bouncing a photon (or graviton?) off it.  Or merely absorbing a photon it just emitted.  Either way, the photon interaction involves the usual conservation of energy, momentum, and angular momentum, and so the values of those parameters change.  Since they are conserved, we see it as "transfer" from one object to the other.
     That our measurements change the measured object is even more apparent when we realize that the photon (or graviton, etc) is the exchange particle in the interaction between ourselves and the object.  When our eye captures a photon, a charged particle in our retina and a charged particle in the object we observe are each exerting a force on the other (Newton's third law), a force "carried" by the photon.  This assures that the observer (his eye) is entangled with the observed.
     Whether the observed object "exists," if unobserved would seem no more a valid question here than in the ancient (pre-quantum) conundrum of "If a tree falls in the forest is there any sound if no one is there to hear it?"  Whether the value of the measured parameter exists and has some declared value is answered by quantum mechanics for measurements of very, very small quantities.  QM says it fluctuates and is statistical and depends on our interaction with it (measurement), and so does not exist in the classical and intuitive sense.  Herein lie aspects of quantum mechanics which can lie well outside the edges of human comprehension.
     Like everything else quantum mechanical, when we go to systems of huge numbers of quantum mechanical particles, we go to a "classical" system—and it's a statistical system—which behaves as our observations have led us to expect.  (The realm our perceptions and intellect evolved to deal with.)  Schrödinger’s cat is a classical cat.  Whether or not the unstable isotope has fissioned and released the cyanide is first a problem for classical statistics.  A person who would write the wave function for the cat, isotope, cyanide and box to try to determine whether the cat is alive or dead would probably write the wave function for the air molecules in the room to try to determine when they will all statistically gather in the corner leaving a vacuum in the room.  However, such interesting problems don't necessarily have humanly interpretable answers.
     Past experience with scientific discovery suggests that the wave-function writer is probably asking a lot of wrong questions.  Nevertheless, the future of science depends on those who keep seeking new questions.
     Better and better questions.