Maybe it's the grouping of conservatives and libertarians together. Maybe that's where my head gets lost.
Take private property as a canonical example. Private property is good from both liberal and conservative perspective, but for different reasons. Liberals like private property because it confers more freedom to individuals, while conservatives like it because of its hereditary nature, as a reward for being good manager. These are two different values. (Oh and BTW, Marx was one of the first people who pointed out this contradiction, if you reward capitalists with property for doing good, eventually you're gonna run out of it..)
People have different definitions of marriage. One person ‘defending’ one definition looks like ‘actively working against’ another definition.
E.g. in some cultures, by definition, a marriage isn’t valid without a male/female sexual act (the only kind that can lead to procreation, by definition). This axiom is held by over a billion humans for example.
Take racial disparity in the US for instance, and the history of racial segregation and racist policies in government institutions. How do we address the inequality present in the current system without first addressing the root causes of that inequality; some of which stem from hundreds of years ago?
By your definition, that would seem to be special pleading, but it is not, in reality. It is attempting to address current conditions that were caused by past conditions. Or am I way off base?
However, I also thinking most advocacy is falling way short of acheving stronger community ties for marginalized groups. Indeed, a lot of such advocacy is clearly counterproductive, in pushing for "liberal" social atomization and further marginalization.
It's quite normal as we grow up we become more conservative, especially successful people - we want to continue the success that was based on values we grew up with. That's why conservativism is associated with belief in authority - you need authority to enforce the values of the past.
So when you grew up, you perceived freedom of speech as a good value because it meant freedom, and now you perceive it as a good value because it was there when you grew up.
However, liberals moved on, and found other issues that concern human freedom, which were neglected before. So for instance, there is freedom to have a different sexual orientation, and be able to express it. Conservatives disagree with that value because it simply wasn't part of our culture in the past.
(Progressives - or perhaps socialists would be more fitting name - are based on another value, namely inclusion in the community. They are distinct from liberals.)