Ah... the good ol' logical fallacy. Correlation does not mean causality.
You can say windmills are turning because there's wind but you can't say there's wind because it was caused by the windmills turning, the causality only has one direction even though there's correlation between both events. In our particular case, the higher the skill the higher the MMR will be but an higher MMR doesn't necessarily mean the player will be more skilled. It will only mean that he won more games than the ones he lost. Hero spammers are the most common and easiest to understand proof of this concept. Just because you can play 1 hero perfectly and gain MMR doing it, it doesn't mean you're highly skilled, all it shows is that you can game the system. There's also the copycat effect, where a lower skilled individual just copies what a higher skilled one does without fully understanding why that thing works and still manages to be successful with it (very common in CCGs for instance).
As for "true MMR" (your expression), I do believe hero/position spammers tend to not be as good as the number indicates simply because they only employ a subset of all the skills in the game or only apply those skills in a very small pool of situations. Is it hard to imagine that a player that plays everything and manages to maintain a steady, albeit slow, increase in MMR is vastly more skilled than a fast climber that only uses fotm heroes/builds?
That's an excellent excuse to post this: