I think what is left out these days is nuance and a judgement of individual actions. Hamas is a horror show, hands down. Israel has a horrific expansionist agenda. It’s not clean and neat. We don’t need to cheer a side on all occasions. Let’s condemn atrocities and organisations with atrocious actions actions (Hamas and Israel) and push for fair and equitable outcomes for the innocent citizens on both sides.
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.