First of all, from the practical / Physics point of view you will see that it actually makes no difference, it's just mathematical details. But I do understand the desire to have a mathematically precisely defined theory.
Observables are in fact required to be self-adjoint, and the reason is - like you guessed - that we need the spectral theorem (see e.g. here).
Since an essentially self-adjoint operator has a unique self-adjoint extension, it does not really matter whether we write down a self-adjoint operator or "just" an essentially self-adjoint one.
In a Physics lecture, the professor will usually only write down the momentum operator as $p = -\mathrm i\, \partial_x$ without specifying a domain, and prove that it is symmetric while implicitly assuming that the wave functions are continuously differentiable or something similar.
Explaining the issue of domains of unbounded operators and introducing Sobolev spaces etc would take a lot of time for arguably little benefit.
The property will then be called "hermitian" which seems to be used in the meaning of "self-adjoint but we don't really care about the details". (As far as I know, hermitian is originally supposed to mean bounded self-adjoint. See also here.)