Well, yes, indeed theoretical physicists tend not to think very much about such subtleties, but from time to time you start nevertheless thinking.
Sometimes you even need to argue with the domain and co-domain of operators and whether an operator is self-adjoint or only Hermitian. It's surprising how simple these questions can be. E.g., when I prepared my lectures about orbital angular momentum for quantum mechanics, I started in the usual way using Dirac's representation free formalism. From the Lie algebra (angular-momentum algebra) you find of course that angular momentum ##\vec{L}^2## takes the values ##\ell(\ell+1)## with ##\ell \in \{0,1/2,1,\ldots \}##. On the other hand "we all know" that orbital angular momenta only have integer values for ##\ell##.
Now, why the heck is this so? Of course, first I started to look in some textbooks. There you find the argument within Schrödinger wave mechanics that when separating the Laplace operator in spherical coordinates you get the eigenfunctions of ##\hat{L}_z=-\mathrm{i} \partial_{\varphi}## to be ##\exp(\mathrm{i} m \varphi)##, and then "##m## must be in ##\mathbb{Z}##, because the wave function must be a unique function of ##\vec{x}##. Thus as function of ##\varphi## ist must be ##2 \pi## periodic and thus ##m \in \mathbb{Z}##." Well, hm, sounds convincing, but than before I carefully told my students that pure states are represented by normalized wave functions and that an overall phase doesn't change the state. The if I make ##m## half-integer, all that happens when checking what happens when changing ##\varphi## to ##\varphi+2 \pi## is that I get a factor ##(-1)##, which is just an unimportant phase factor! So this argument is hand-waving and a bit dishonest towards the students, and of course you always hope that they'll catch you on such a hand-waving "demonstration".
Now fortunately there's a nice way to write ##\hat{L}_z## in terms of annihilation and creation operators of the 2D harmonic oscillator in the ##xy##-plane, and this leads indeed to ##m \in \mathbb{Z}##.
On the other hand, there should also be an argument within Schrödinger's wave mechanics, and there the argument that ##\hat{L}_j## should be self-adjoint and not only Hermitian plays an important role! Indeed you can just bravely trying to find spherical harmonics with ##\ell## and ##m## half-integer. Checking ##\ell=1/2## is already sufficient. You easily find the solutions for ##m=\pm 1/2##, looking innocent enough, being even square integrable on the unit sphere! But then check that applying ##\hat{L}_-## to the ##m=1/2## solution gives the solution for ##m=-1/2##, and this indeed fails! You also don't get ##0## when applying ##\hat{L}_-## to the solution for ##m=-1/2##, but according to the abstract Dirac formalism this should be so! Now indeed, the solution of this apparent paradox is that the solutions for ##m = \pm 1/2## are not in the domain of ##\vec{\hat{L}}## as self-adjoint operators, i.e., their repeated application to this apparent solutions lead out of this domain, and thus you cannot realize the case ##\ell=1/2## as square-integrable functions on the unit sphere, and thus you have to abandon the idea that there were an orbital angular momentum with ##\ell=1/2##. That really shows that spin 1/2 can only be unerstood within QT, and it's not simply some "canonical quantization" of classical physics.
Another related question is, what is the representation of an angle in quantum theory. Naively you'd expect it's the canonical conjugate variable to an angular-momentum component (as in classical mechanics with the analogy between commutators in QT and Poisson brackets in CT in mind). Then you immediately also run into a contradiction, because then from the treatment of linear momentum and position operators you know that then both should have a continuous spectrum, and it's entire ##\mathbb{R}##, but that contradicts the angular-momentum algebra proof that ##\hat{L}_z## has only ##\mathbb{Z}##, which we had just shown before with all the quibbles to throw away the false half-integer eigenvalues. Here the solution is simple: Knowing from the analysis leading to this result, that it is indeed true, that a rotation by ##2 \pi## indeed should be represented by a phase factor ##+1## when applied to the wave function, we should have the wave functions ##2 \pi##-periodic wrt. the azimuthal angle ##\varphi##. The naive realization as the canonical conjugate variable would be of course to multiply the wave function with ##\varphi##, but starting with a ##2 \pi ##-periodic function and then applying ##\hat{\varphi}## leads out of this function space, since ##\varphi \psi## wan't be ##2\pi## periodic anymore, and thus ##\hat{\varphi}## is not self-adjoint, because the domain and the codomain should coincide for this.
So finally, what can be used as a representation of an angle are the functions ##\cos \hat{\varphi}## and ##\sin \hat{\varphi}## (defining uniquely a point on the unit circle) or even the unitary operator ##\exp(\mathrm{i} \hat{\varphi})## with ##\hat{\varphi}## defined in the "naive sense" as multiplying the position or momentum wave function by ##\varphi## or rather the functions thereof.