This is wildly wrong. The only thing you got right is that the MAX was designed to avoid needing a new type certification.
Boeing radically shifted the Cg to accommodate changing trends in the commercial airline industry which were resulting in Airbus's planes being more competitive. They shifted it well outside what would be considered prudent.
This required flight control software bandaids to keep the plane stable (or perhaps better put, from entering a flight envelope where loss of control would happen), but they then also cheaped out of providing redundancy for some critical sensors... AND cockpit annunciators to alert on failures of the sensors.
Boeing then passed off the Cg shift and additional systems as very minor changes in the filing for their airworthiness certifications, the FAA rubber-stamped everything because they didn't have the staff to review all these certificate filings so they largely trust manufacturers...
...and then making things even worse, Boeing did very little to disclose the additional stability system and sensors to the airline's chief pilots who develop the in-house training. To say they were furious to learn they hadn't been told of the new systems would be an understatement.
The MAX crashes would have been substantially less likely if the pilots had benefitted from their airline's chief pilots knowing of the safety systems and having training on how to handle failurs, as well as warning annunciators for sensor failures....but Boeing wanted to save a thousand bucks or so per plane - or more likely, wanted to avoid drawing attention to the situation and having people ask "why is a critical flight control system dependent upon a single sensor?", which is likely why they also did little to inform airlines of the new systems, train the chief pilots on it, etc.
And I am not saying that the software bandaid and the goal of avoiding a new type certification is a bad thing. In fact, it is done all the time with no ill effect.
What I am saying is that they did it wrong, and that's the real problem. Your explanation on what they did wrong is, I think, entirely correct.
My hypothesis is that if they didn't care about keeping the same type rating, they probably could have skipped the bandaid and trained pilots to just keep out of that dangerous configuration, just like pilots are trained to recognize, avoid and recover from stalls. It didn't look like an unmanageable problem for pilots, more like something they needed to be aware of and train for.
There is a thing called "type rating", that is qualification pilots need to have in order to fly a specific type of plane. For example, pilots need a type rating for the Boeing 737 in order to fly a Boeing 737.
Manufacturers try to do everything they can not to require another type rating when they make changes to an existing aircraft, because type ratings require expensive and time-consuming training that pilots and their companies would rather avoid, and it influences purchase decisions. One of the root causes of the B737-MAX crashes is that Boeing tweaked some settings in order to make the B737-MAX behave like the original B737 when it was, in fact, a different plane, and they did it wrong.