Does a home warranty or insurance cover AC repair?

Two different things get confused here. A home warranty is a service contract you pay for annually that covers repair of systems that fail from normal wear, and most of them do cover the AC, subject to a service fee per call, coverage limits, and exclusions for systems that were not maintained or were improperly installed. The common frustrations are the warranty company sending its own contractor rather than yours, slow scheduling in peak season, and a cap that does not reach a full replacement.

Homeowners insurance is different. It covers sudden damage from a covered event: a lightning strike, a tree on the condenser, a power surge in some policies, a fire. It does not cover a compressor that wore out or a coil that corroded. If a storm did damage the unit, photograph it before anything is touched and ask the contractor for a written report that says what the storm did, as distinct from what age did.

Whichever applies, the diagnosis is the document that decides it. A contractor who writes down what failed and why is giving you what either company will ask for.