I'm definitely a relentless door keeper, it'd bother me to let the doors close.
I think there may be credence to this theory, but it'd be hard for to make such a drastic shift in personality after years of the "don't burn any bridges" method.
Frankly, I've had very good things come from doors that I could've easily let close but didn't, so I have a volume of anecdotal experience telling me that I shouldn't burn any bridges, lest I need them in the future.
I'll have to think on this one...