Whilst I agree with your conclusions I am conflicted about how you define “problem solving”.
I would think most people agree that a process-oriented workflow will always yield better results. However, the lack of process or methodology does not come from a problem-solving mindset.
We solve problems on a daily basis. Coal power is a (now out-dated) solution that provided energy for centuries and enabled the Industrial Revolution. Even that primitive, polluting technology was iterated over the course of the centuries.
Just because it is unlikely we will build the one monolithic product that withstands millennia of environmental and behavioural change doesn’t mean we shouldn’t look at problems and try to alleviate with them.
Problem-solving is not a singular event. It is a constant, iterative balancing of trade-offs.
And by that definition, we should all absolutely be solving problems. From the cleaning lady to the CEO.