To echo what @maxholloway and @Millie said I doubt that a staking module is an effective measure to backstop insolvencies. Accumulating large stablecoin reserves is by far the best defence against such an event.
I agree that a Balancer pool type staking module like Aave could have the added benefit of increasing EUL liquidity. But I would argue that it is probably not the best use of the team time to focus on this when other options like adding EUL liquidity directly might be simpler and just as effective.