Ich kümmere mich um meine Familie.

English Translation

I care about my family.

Is “I care about my family” actually a good translation? Isn’t the meaning closer to “I look after my family” with “kümmern sich” rather than “I care about them”?

You’re right, it’s a bad translation. And you’re correct with your analysis:
sich kümmern um - to look after / to care for (not about)
to care about - hard to translate. maybe “Meine Familie ist mir wichtig” or “… nicht egal”.