In my quest to understand quantum theory and find a reasonable interpretation
I recently came across this paper (pdf!) about 'The Impossibility of Accurate State Self-Measurements' by Thomas Breuer.
It is very similar to how I think about all this.
"It is shown that it is impossible for an observer to distinguish all
present states of a system in which he or she is contained ...
[..]
As far as quantum mechanics is concerned, von Neumann assumed the theory
to be universally valid and thus was led to the measurement problem.
Bohr denied the universal validity of quantum mechanics for 'purely logical reasons',
and thereby avoided confrontation with the measurement problem.
It has often ... been suggested that self-reference problems for universally valid
theories may pose serious difficulties for a quantum mechanical description of
the measurement apparatus. The aim of this paper is to investigate these suggestions."