No, the video does not imply that the uncertainty principle is false.
But it does imply that you have to be careful/precise to claim what the uncertainty principle is about. It is actually a claim about the wavefunction. In many interpretations of quantum mechanics, we equate that to a claim about the properties of particles. However, the latter is dependent on one's interpretation of quantum theory. The video talks about the de Broglie-Bohm (or pilot-wave) interpretation of quantum theory. In that interpretation, there are two separate quantities: the wavefunction, and the particle. At a fundamental level in that interpretation, the 'uncertainty principle' applies to the former, not to the latter (since the particles have well-defined position and velocity).
That being said, the property that makes the de Broglie-Bohm interpretation consistent with the observations of quantum theory, is that one presumes one is in 'quantum equilibrium', where one equates one's (classical-like) ignorance of the particle with properties of the wavefunction (i.e., Born's rule). In fact, the de Broglie-Bohm interpretation is itself actually an umbrella term for two distinct intepretations: one where this quantum equilibrium is one of the postulates, and another where it is derived as a statistical property/consequence (similar to how in classical statistical mechanics we derive that a classical gas is described by a homogeneous distribution in classical equilibrium). I do not personally work on these topics, so I do not know how one justifies making classical ignorance part of one of the fundamental axioms; seems strange to me.
