Before the late 1960's, there were in fact two competing interpratations of the Heisenberg uncertainty principle. The 'realist' school believed, as you do, that a particle does have a definite position and momentum, but that they are unknowable (or at least not predicted by quantum theory). The competing 'Copenhagen' interpretation, which has become the accepted one, holds that a particle does not have a definite position or momentum until a measurement is taken.
Believe it or not, this is a testable proposition. In 1965, John Bell proposed an experiment to rule the "hidden variable theory" in or out. Wikipedia explains
here and
here.
Repeated experiments have shown that the universe violates Bell's inequality and therefore does exhibit randomness.