It's not over-diluting that could make it "ropey", it's over-chilling. Try putting a bottle of vodka in a freezer. As you chill spirits their viscosity increases. Shaking high ABV drinks with ice brings their temperature well below 0C. See:
Stirring is much less effective than shaking, but very long stirring could approach shaking temperatures. I think it's plausible that the viscosity difference could be perceptible.
Well when you shake it the viscosity difference isn't noticeable. A shaken martini sucks but I wouldn't call it "ropey".
They aren't putting it in the freezer, they're stirring. As that link you posted shows, stirring for 7 minutes and shaking probably get about the same dilution/chilling.
http://www.cookingissues.com/index.html%3Fp=4585.html
Stirring is much less effective than shaking, but very long stirring could approach shaking temperatures. I think it's plausible that the viscosity difference could be perceptible.