There is no easy answer to the question of whether damage caused by an above ground pool is covered by your home insurance. Read your policy carefully, looking for exclusions concerning pools. If you are unable to find it in your policy, contact your insurance company and have them tell you where it can be found.
This scenario does not apply to swimming pools alone. It could just as easily be asking about a water heater, washing machine or dishwasher. A swimming pool is not an appliance, but the effects of it collapsing fall into the same category of water damage.
If there is no provision for above ground pools in the wording of your policy, it is probably included under structures on your property. Water damage from many household items is covered, but a swimming pool is sometimes considered a much higher risk, so additional coverage may be required. The damage would not be considered flooding, because the damage would the result of a device not functioning properly, rather than water that rises outside of its boundaries. If your agent tries to tell you otherwise, request a review of the case for further clarification.
In some policies, above ground pools may be excluded from coverage completely. In that situation, you can buy a separate policy for pool related damage. That coverage can be added to your home insurance, and it would also be a good idea to add liability coverage for the pool as well. Pools can be exceptional damaging, and many companies will require you to fence around it before they will write a home insurance policy on the home.
As you can see, there are many different variables that will decide whether damage caused by your above ground pool will be covered. Your homeowners insurance is not exactly like the policy covering your neighbor's or someone down the street. The only way to know exactly what is covered and what is not is to become familiar with the way your policy is worded.