Let’s remember our definitions, conservatives like things as they are now or were at some earlier time. Progressives believe things as now structured are wrong and need changing.
Perhaps it is the other way around? What if happiness causes conservatism instead of conservatism causing happiness? Perhaps unhappiness causes liberalism/progressivism - the desire for change?
Maybe being happier is more likely to make you conservative? After all, if you are happy your life is good, this goodness changed circumstances might well threaten.
For unhappy people, life and its circumstances are an uneasy fit and outcomes leave much to be desired. Instead of taking blame for their own unhappiness, they blame it on external factors which from their point of view need to change.
For most people in our society their life outcomes are roughly proportional to their accomplishments, more and better inputs lead to more and better results. Those who make good choices have better outcomes than those who make poor choices.
And then there is a small but highly visible group whom fortune has favored far beyond their reasonable expectations. Who ended up rich beyond the dreams of avarice, and who feel guilty about their undeserved gains.
Having experienced the capriciousness of fate, they fear maybe the poor and downtrodden are its victims. To atone they end up favoring change too.