Expected value in usual quantum mechanics vs quantum information

In standard Quantum Mechanics, one computes the expected value of an operator $$A$$ (arbitrary state $$|\Psi\rangle$$) as

$$\langle\Psi|A|\Psi\rangle.$$

This has the virtue that we can compute for instance the average energy of the system $$E=\langle \Psi|H|\Psi\rangle$$ and also we can compute, if we want, the expected value of non-Hermitian operators like $$\alpha = \langle \Psi|a|\Psi\rangle$$ in the Harmonic oscillator. Now, in Quantum Information or Quantum Open Systems, one usually works with a complete set of measurement (Kraus) operators $$\{ K_m\}$$ that fulfill

$$\sum_m K^\dagger_m K_m=\mathbb{I}.$$

Then, for instance, the majority of quantities that we can measure (if not all) will be Hermitian, since for instance

$$\langle\Psi|a^\dagger a|\Psi\rangle.$$

On the other hand, as it is true that e.g. $$a$$ is not a Hermitian operator, in phase space techniques one manipulates such quantities theoretically and they are very useful for studying the system dynamics.

My questions are: can the set of measurement operators $$\lbrace K_m\rbrace$$ can at some point accommodate such non-Hermitian quantities? Why have a formalism that in principle "loses" a big amount of (theoretically) useful information?

You can measure the expectation value of any operator $$O$$ if you can measure hermitian operators, by decomposing it as $$O=H+iA\ ,$$ where $$H=(O+O^\dagger)/2$$ and A=$$(O-O^\dagger)/2i$$ are both hermitian, and measuring the expectation value of $$H$$ and $$A$$ individually.
In particular, for the POVM measurement formalism you describe above (what you term "Kraus operators"), given the eigenvalue decompositions of $$H$$ and $$A$$, $$H=\sum_{i=1}^n h_i|H_i\rangle\langle H_i|\,,\ A=\sum_{i=1}^n a_i|A_i\rangle\langle A_i|\,,$$ you can define a POVM measurement with operators $$\{K_k\}=\{\tfrac{1}{\sqrt{2}}|H_1\rangle\langle H_1|,\ldots,\tfrac{1}{\sqrt{2}}|H_n\rangle\langle H_n|, \tfrac{1}{\sqrt{2}}|A_1\rangle\langle A_1|,\ldots,\tfrac{1}{\sqrt{2}}|A_n\rangle\langle A_n|\}\ .$$ Given the outcome probabilities $$p_k$$, you can then compute $$\langle O\rangle$$ as $$\langle O \rangle = \sum_{k=1}^n (p_k h_k + i p_{k+n} a_k)\ .$$
Finally, note that whenever $$O$$ is unitarily diagonalizable (but not necessarily hermitian - namely, this holds if and only if $$O$$ is normal), you can compute $$\langle O\rangle$$ by defining POVM operators $$K_k$$ as the projectors onto the eigenvectors of $$O$$, as above.