The short-term case for other banks bailing out depositors is that any systemic risk affects all banks and the rest of the citizenry, albeit indirectly.
But I think it's probably economically sound in the long-term as well. That is, asking every depositor with $250K+ to assess the financial health of each bank they use and maybe buy insurance is collectively more expensive than just having the FDIC implicitly insure all deposits.
That's different than asking if it's "fair". But I would wager it's probably more efficient.
Insurance always has a cap in the payout at some dollar amount, usually directly related to the amount one is willing to pay for the insurance. It's usually up to the insured to balance those two factors to get adequate, but not excessive, insurance for the risks they are exposed to.
Most of the arguments I've seen are effectively arguing that 250K is too low an amount. While that may be true, that was the well-established 'rule of the game'. The FDIC limit was no doubt chosen, as most insured amounts are, to cover the majority of damaged parties, for an acceptable cost.
This isn't grandma or Joe/Jane Public losing their life savings; the FDIC insurance easily covers the vast majority of individuals depositing cash. These are businesses that have, or should have, the financial wherewithal and resources to mitigate their risk beyond the FDIC baseline.
But I think it's probably economically sound in the long-term as well. That is, asking every depositor with $250K+ to assess the financial health of each bank they use and maybe buy insurance is collectively more expensive than just having the FDIC implicitly insure all deposits.
That's different than asking if it's "fair". But I would wager it's probably more efficient.