It may be cheaper to replace screens than lose the cage (the pool/patio enclosure frame.) But since South Florida hasn't had a hurricane since Wilma in 2005, this procedure hasn't been tested.

Hurricane Irma’s effect on South Florida’s vibrant real estate market is not likely to be much more than the hassles ... debris in the streets and minor damage to screened-patio enclosures, ...

Almost all screened or three-season patio rooms can be built using a concrete patio as a base. However, many patios are not built with deep footers.