This concept of "caring" is just a new coat of paint on "we're not just a workplace, we're a family," a strategy to get your employees to volunteer for more responsibility in exchange for kudos (rather than the pay or title they would demand if asked directly).
You don't need to "volunteer" anything, or do unpaid labour—to care. All this attitude of "the world/company/country owes me big-time" is poisoning your soul, man.
It's not about entitlement, is about healthy boundaries around working life. There is no virtue in burning yourself out. Your manager who tells you that you're "part of the family" is not being honest with you. When your interests and your employer's interests diverge, it will turn out that it was just a workplace, after all.
Ask your coworkers who "care" what they think.
