A (very) hypothetical conservative party not biased in favor of the rich over everyone else

I've wondered how the Republican Party or conservative parties elsewhere could avoid being the party that favors the rich over everyone else, especially the poor. As long as we have progressive taxation, or even a flat income tax that is not a poll tax, then the whole "smaller government versus larger government" dimension biases the conservative party towards the rich, whose economic interest at a simplistic level favors smaller government. So the libertarian angle that's being pushed now isn't one that's going to change the class favoritism.

Fighting and slowing social transformation, the other side of conservatism, doesn't map immediately to class issues. You might even expect the radically-increasing economic inequality today compared to past generations to disturb some social conservatives. I think the rare and usually-unsuccessful efforts by some conservatives to alter anti-tax positions at state levels might stem in part from social conservative viewpoints. On the other hand, social transformation has generally involved acceptance of outsider groups that experience economic discrimination among other harms, so fighting attempts to fix this often aligns with being biased in favor of the rich over others.

A dimension that doesn't divide parties right now in the US is opportunity versus compassion. One party could favor spending government money, whatever the total amount may be, on funding sufficient equality of opportunity for the vast majority of citizens to achieve what they want with their lives. The other party could favor spending government money, whatever the total amount may be, on compassionate assistance to people regardless of whether they share some of the responsibility for their problems. The first party seems to me to be logically right of center, and the second to be left of center. The Opportunity Party would also tend to be the party of the young and the Compassion Party that of the older demographics. This is the opposite of the actual situation in the US where younger people are more Democratic-leaning relative to older people.

So that's the first and biggest problem - Republicans aren't very willing or able to switch generational sides in a conflict over resources. The other issue is that Republicans aren't really competing for either the Opportunity or Compassion label. Instead they hug tightly to the Small Government label, leaving both of the other two to the Democrats. The recent noise from the right about achieving equality of opportunity through the destruction of the liberal welfare state is just that - noise. If they start putting more resources behind achieving equality of opportunity than Democrats are willing to do, only then will things start getting interesting. Unlikely.