Two different policies, two different jobs
The complex insures the pool, the fence and its own responsibility for maintaining them. Your personal liability covers what you are responsible for: a guest you brought, a child you supervised, something you did. Tenants assume the pool being theirs to use means it is somebody else’s to insure, and only half of that is true.
Guests you bring are your exposure
Inviting friends to use the amenities makes you responsible in a way that simply living there does not. An injury to a guest is a liability claim against you, and the limit on a renters policy bought to satisfy a lease is usually the minimum one.
Grills and balconies are the other common claim
Fire from a balcony grill spreads to other units and the resulting claim can be enormous relative to the cause. Many complexes prohibit them for exactly this reason, and a prohibited activity is a difficult position to defend.