For me political conservatives recognise the value of many of our establishments and don’t want to see radical, untested, change risk what we have.
Progressives want to see us improve our systems, institutions, and cultural norms. Laudable goals, however the goals and implementation approaches can produce results that diminish, rather than enhance, what we already have.
Socially conservative values, to me, are another kettle of fish. I know socially progressive conservatives and socially regressive progressives.
Libertarianism is also a separate, cross over, category for me. How power is shared between individuals and government is a critical foundation for how we function and can once again be adopted by politically conservative or progressive ideas.
Disclaimers:
Not American, Libertarian leaning Socially progressive Somewhere between a political conservative and a political progressive
Conserving doesn’t mean reducing, it means protecting what already exists. Restricting access to birth control, for example, I would not call politically conservative.
Republicans have, at different points in time, been fairly forward leaning in their own way. Have a google of progressive policies implemented by republicans.
Social conservatism is more akin to what you are describing rather than the conservative political movement (and has a large section of the republicans in its sway at the moment). The thing they are trying to protect is their own personal moral view of the world and in so doing they are willing to not conserve the current political system to achieve this goal.