I can’t imagine anyone is fooled when a Democrat lies about his beliefs. Everyone knows what he believes.
The purpose it serves is to upset opposition attacks, is all. Conservatives know he believes in gay marriage, indeed, he’s probably gay himself. His supporters know he believes in it, and that he is only lying for tactical reasons, and they are fine with that.
I’ve known Democrats who, when you point out a lie by their favorite guy, will quite openly say “of course he says that, he has to...”
Democrats admired Clinton for his ability to spin anything. And they admire Obama for the same reason. They don’t demand honesty from their politicians, they demand that he validate their prejudices. If he’ll do that, they’ll follow him all the way to hell.
Your closing sentence describes the majority of democRat voters with glaring, unvarnished accuracy. There are several of that exact species among my relatives; by marriage, of course - and also some acquaintances.