Way back in the beginning of the second half of the last century Francis Schaeffer answered this question by showing how a wall had been constructed by man between man and God. He cautioned that the new way of looking at man was as machines. This morphing of man into machine was the result of a strictly material view of man; there is no consciousness aside from that created by the cells and electrical currents in our heads; no soul in need of salvation; only material. With this view man necessarily is only a very sophisticated machine whose actions and decisions are determined by the programming that is the result of millions of years of survival.
This materialistic view of the world found itself in opposition to religion. As a result of the deconstruction of all thinking to preprogrammed electrical currents religion, along with the thinking that life had some eternal meaning beyond surviving for the purposes of consuming copulating and excreting, was expected to subjugate itself to this new thinking. To the extent that Wallis and Currie have been successful, this subjugation has been successful. As the mind of man has effectively closed all knowledge except that gained from empirical evidence has been rejected as "unknowable". "unknowable" is key to understanding the new "post" modern view of religion. More on this latter.
So what is the fallout of this new expectation of religion? The embracing and acceptance of religions that subject themselves to this new view of religion in general proves that the idea of religion is too tenacious to be eradicated from the minds of modern man even in the most materialistic of societies. It appears that upwards of ninety percent of all people are not willing to embrace their own meaninglessness. This brought about two results.
First, a truce was effected between modernism and religion. Modernism set forth the boundaries within which religion had to function and religion agreed to be irrelevant in exchange. Beyond the Prozac effect that comes with the creating of ones own god willing to subject itself/herself/himself to modernism's terms and conditions, religion lost its impact on culture. All western churches are to some degree held captive to these terms and conditions as can be seen by the extreme focus on temporal material, comfort and inner-peace, and the diminished focus on things eternal. Western churches therefore, to the extent they have knowingly or unknowingly bought into this new way of thinking, have been reduced to a Prozac of sorts for the soul. A feminized version of God: He loves you very much, He knows what you're going through, and He cares deeply about you is the central theme it seems these days, a theme that is used as a salve for our deep sense of meaninglessness. Gone are concepts of man's depravity, Hell, God's wrath, and the so-great-of-a-Salvation accomplished on the cross; these are not salve; they are not Prozac, they rightly provide no inner peace for the soul, but they are just as true.
Secondly, religion in general has been used quite effectively in history as a means of accessing power. This is where Christianity stands apart and why Christianity, as opposed to other religions, is attacked daily, and attacked no less from a position of proclaimed ignorance, read we-can't-know. Assertions of truth are made against the unknowable from this position of we-can't-know. But the attacks are not against those that have set of house keeping this side of the we-can't-know Wall like Wallis and Currie. No, the attacks are directed against those who refuse to mind that Wall and thereby deny those who embrace religion for the purpose of access to power the power they desire. In short, it denies political hacks posing as "reverends" of control over the thought life of Christians and their vote. How funny it is that this seems to be the number one complaint against those who have been termed by those who lust for power beyond their grasp using the language of power: "the religious right".
So how do we use this narrative to answer the question posed by the Title: "Which Jesus do they worship?", and Neil's question: "Do they really care what Jesus would do?". In my mind the answer to the first question answers the second. Schaeffer points out in "How Then Should We Live" that since a Wall was erected by man between God and man, anything put on the other side of that wall is only a symbol representing what man desires of the god he has created. This changes our language into babble, for the word Jesus no longer means the same thing to all who use it. For some, like the Wallis/Currie camp who see religion as an access point for political power, the word is symbolic of their Marxist/socialist/communist/leftest/atheistic/material centered redistributive political aspirations. So by pointing out that the Jesus they worship is only a word: "Jesus", that symbolizes anything but the Jesus that was born of a virgin, raised people from the dead, died on the cross to save from the wrath of a Holy and Righteous God those who deserve that wrath--all of which are an anathema for the materialist--we know that it is a different Jesus. So by being a different Jesus; a Jesus conflated into a symbol representing the utopian desires of depraved men, we can truly answer Neil's question with "Yes", they do.