Yes, you can reject Bohr's model on the grounds of the uncertainty principle alone.
Bohr's model assumes precise values of orbital radius and velocity. (Hence you can work out the energy (KE + PE) differences between orbits, giving the spectral transitions.)
However, the uncertainty principle explicitly forbids this (can't simultaneously know exact values of an electron's position and momentum).
Let's review what a model is. It is an incomplete representation of the real thing so as to capture how some limited aspects of the real thing behave or appear. That's all a model is. By its very nature (and definition) a model, any model, is wrong in some way... always.
The Bohr model, applied to the hydrogen atom is a perfectly good model. It gives results consistent with observations. So it's OK to apply the Bohr model to problems addressing the hydrogen atom and its orbiting electron.
But, as you and gin noted, we would be wrong (not the model) to apply the Bohr model to other, more complex elements. This is where quantum models, e.g., the orbitals rather than orbits, must be invoked. And of course, wherever quantum physics is invoked, the HUP reigns.
The Bohr model yields extremely accurate results for the hydrogen atom. We know this from the measured emission spectrum, and from the theoretical emission spectrum matching. This implies that we shouldn't at all try to reject Bohr's atomic model completely. There is STILL some truth to it.
Even other atoms sort of match the Bohr model, but it is harder to know, because of electron-electron interactions.
The difficulty with Niels Bohr Hydrogen model energy levels is that he assumed that the electron mass would not change irrespective of its velocity. Hence he did not give a true picture of what was happening in the atom.This was in disagreement with Einstein equation of mass change with velocity.
The equation would have some validity if it is assumed that the momentum of the electron in the Hydrogen atom is really a constant.
Heisenberg Uncertainty principle assumed a moving mass at any velocity would not change but would have different momentum if its velocity changed.
However he stipulated that it is impossible to determine velocity of a moving electron of a constant mass and Its position simultaneously.
Hence as per the Uncertainty principle Velocity is the ratio of a displacement and time.
Displacement indicates postion which can be measured but as soon as you measure position time has alreadiy changed. Thue we cannot measure position and time instantaneoulsy.Thus you cannot measure the momentum of the electron and its position at the same time.
The falacies in both Heisenberg and Niels Bohr and Eisntein is that they all left out temperature as the factor that causes mass to change in the electrron .
The famous equation E=MC^2 that Einstein used to determine the energy content of the electron should have been changed as per the following equation;
E(t) = M(t) C^2
Where E(t) is the Energy content of the electron as a function of temprature
M(t) is the mass content change of the electron as a function of temperature
C^2 is the speed of light square in local space.
hence they all missed the obvious which is temperature.
Thus what we assume what we know is correct is not always necessarely so.