I'm in aviation claims, mostly on the GA side but have been involved on the airline side. In general, American carriers have massive limits, 1 billion and above. These types of policies are "vertical". Which means several insurance companies get together and divvy up the risk. 5 or 6 companies take a share, adding up to 100%. The premium, and the claims payments are divided per the agreement. There is a "lead" insurer, who manages the claims on behalf of the following co-insurers.
In the case of a major loss with multiple fatalities, everyone will be sued, including various OEMs. It gets very complicated. Depending on the cause of the accident the airline can sue an OEM, co-defendants can work out settlements before trial, etc. Insurance pays for legal defense so there will be several firms defending depending on the where the suits are brought, and of course the airline legal counsel is involved.
Then insurers back themselves up with reinsurance, which is insurance for themselves. Which is a who other world.