No, it doesn't. Read more carefully. Wider handlebars mean you have more leverage over the front wheel. That's it. They don't make the bike "more twitchy." They allow easier resistance to unintended or external control inputs, like wind, or the inertia of a load on the bike, etc. Absent those factors, which are important, bar width isn't relevant to handling and having wider bars means an unintentional pull on the bar has a greater effect. The front-end geometry of a bike is what determines its handling, everything else is secondary at best.

See above. The confusion here is about what is meant by "stability." Call me crazy, but I consider stability to be a relative lack of response to steering input - being able to do a madison sling means you're very stable. What you're referring to with mountain bike bars isn't about what I would consider stability, it's about fine control and high leverage over the front wheel.