The postulates are roughly quantum system evolve unitarily except when they are measured, then a non-unitary wave function collapse happens. That is inconsistent. For the system under investigation this works, it evolves unitarily until measured, then the wave function collapses non-unitarily. The combined system of system under investigation and measurement device however is never measured, therefore must evolve unitarily. This is a conflict, the combined system can not evolve unitarily and also have some subsystem, the system under investigation, evolve non-unitarily, i.e. undergo a wave function collapse. This problem has been recognized for the better part of a century, that is nothing I am pulling out of thin air.

I mean, it's objectively not inconsistent. If you take the formal postulates you will not produce an inconsistent result.

It is objectively inconsistent, the postulates contradict each other. But for almost a century we did not make any substantial progress, neither resolving the issue theoretically or philosophically, nor experimentally demonstrating any problems. Or at least nothing reached broad consensus.

It's objectively consistent since the postulates do not contradict each other

This seems debatable, but let us assume they are not obviously inconsistent. Those postulates only really mean something if you apply them in some situation, and if you apply them as laid out a few comments before, they make inconsistent claims. If applied to the subsystems, they postulate non-unitary evolution of the combined system, if applied to the combined system, they postulate unitary evolution. If inconsistent is to strong of a claim, I guess this could be changed to incomplete, maybe there are missing postulates resolving the issue.