People can accept that measurements and observables are operators which act on the wave function (which can be expressed as a linear combination of eigenstates of the operator in question). People can even accept that some observables cannot be measured simultaneously, because they do not commute, and cannot share eigenstates.
People cannot accept that particles can only be found in eigenstates of the observable in question, without some mechanism to explain why this happens. If a particle's measured value can only be one of the eigenvalues of the operator, it begs the question "which eigenvalue is it going to show us?". Alternatively, if nature let us measure the expectation value instead of "one of the eigenvalues", then quantum mechanics would not be "weird" at all. Too bad, nature isn't like that, or maybe, be thankful that it is.
> But if the corrections to quantum mechanics represented by the new terms in the Lindblad equations (expressed as energies) were as large as one part in a hundred million billion of the energy difference of the atomic states used in the clock, this precision would have been quite lost. The new terms must therefore be even smaller than this.
I always see people (physicists) complaining about String Theory, etc. because they play around in regimes which are too small for us to actually work with. We have observed macroscopic effects which are due to a build-up of quantum mechanical effects (superconductivity, solid state, etc.). It would be expected that a correction to quantum mechanics would also make macroscopic predictions...