Thanks for that clarification. I have come across the tern contextual in the context of QM and had a rough understanding of it, but this has given me a clearer understanding of the term.Except that there is evidence that there is no such model. What you are describing is a non-contextual model. Which is one which is observation independent. You could say MWI is such a model, I guess, although all possibilities are realized with that rather than none. Not sure that actually solves your quandary.
Past that, pretty much everything is contextual. Which is precisely the opposite - the property & value only come into focus when the measurement is performed.
If there is a quantum system prior to measurement but our models do not or cannot describe it prior to measurement, does that mean that our models are necessarily incomplete? Incomplete models of the system/nature, I mean.