The problem was not originated in their insurance business, but in its much smaller investment banking division, which wrote billions of dollars worth of CDS (Credit Default Swaps), becoming by far the largest counterparty to the whole financial system, allowing banks to reduce their capital provisions on risky loans. You cannot transfer such obligations to other players, and if AIG failed, it would trigger default under such contracts and would seriously errode the already thin capital base of many banks, making things much worse.