You've created a false dichotomy. There is no need to trivialize shared treasures or heritage in pursuit of any cause in order to save anything or anyone. You've decided in some Machiavellian twist that whatever cause you think is truly just is more important than anything other people might value.
It is absolutely important to protect our future, ourselves, and the life we share the planet with, but not by throwing tantrums with unrelated collateral damage. Fight for the climate by fighting for the climate, not by desecrating churches/monuments/art/nature in some weird plight to accidentally piss off the right people and get more TV time.