quantum mechanics - Is it really right to say that we never measure anything exactly in QM?

In reference to this elaborate answer by @DanielSank, I would like to pose the following question(s) in order to verify my understanding of the subject matter--in particular, that of the nature of uncertainty and measurements in quantum mechanics. For context, I would like to quote the most relevant paragraph (most relevant with respect to my question) from the linked answer:

The crucial thing here is that you never ever measure anything to infinite accuracy, so the wave functions resulting from your measurement are not exact eigen-states of what you think is the measurement operator. This is not just "experimental dirtiness". This is a fundamentally important aspect of QM which you should keep near your mental centre as you learn more.

Now, in my understanding, we never ever make a measurement that can produce an exact eigenstate of, say, the momentum operator because there is no normalizable (and thus, physically realizable) eigenstate of the momentum operator. And that, in turn, is the reason why we can never ever measure the momentum precisely--no matter how arbitrarily small we can make the error-bars nonetheless. Any attempt at measuring the momentum will never reduce the given state to a momentum eigenstate but only to a superposition of multiple momentum eigenstates--no matter how closely distributed (i.e. spiked near some particular eigenstate) those contributing eigenstates might be. Thus, in the measurement of the momentum, we will always be endowed with an uncertainty of the scale of the width of that spiked superposition of momentum eigenstates.

Now, all of that is fine and fundamental and not a result of the experimental dirtiness--but in the cases where there are physically realizable eigenstates, any inability of a ''measurement'' to produce a true eigenstate must be credited to the experimental dirtiness--I think. For example, for a Hermitian operator with discrete non-generate spectrum, there seems to be no way for a measurement to produce anything but a specific true eigenstate. In such a case, there would certainly be an indeterminacy in the outcome of the measurement when one starts with a generic state which can be a superposition of more than one eigenstates (and this would lead to an uncertainty in the measurement in the sense that when we are measuring the same observable over an ensemble of identically prepared states, we will not get the same value because different measurements would pick out different eigenstates--generically speaking) but this indeterminacy (and resulting uncertainty of the kind I described) is different from the uncertainty in the measurement of an observable corresponding to an operator whose eigenstates are simply not physical. In particular, once the measurement is made on this kind of an operator which admits discrete non-degenerate eigenstates, all the subsequent measurements are guaranteed to yield the exact same outcome with literally $100\%$ probability except for the experimental dirtiness. And this is the reason I think it is not quite right to say that "you never ever measure anything to infinite accuracy" in quantum mechanics (for example, we always measure the exact spin of an electron to my understanding).

Thus, my core question is that is it correct that it is not right to say that we never ever measure anything to infinite accuracy in quantum mechanics? Or at least can we assert that there are certain operators for which any uncertainty in the measurement must be credited to the experimental dirtiness?