> Insurance only pays out . . . if a bank fails

That's a good point. So one difference is that while the money is equally insured in both cases, the payout dynamics would change. Very roughly, the amount of a payout might be expected to go down in the cross-bank case (smaller account values, but then also more accounts per bank, so it isn't quite so simple), and the likelihood of a payout might be expected to go up (higher chance of failure with more and smaller banks). But this all depends on how interlocked the banks become; in the extreme they could end up functionally a single bank.

The first thing that came to mind for me is somewhat related: Spreading deposits across banks is relatively better for small banks and worse for big ones, since the small banks gain deposits and the big banks lose them. So you can definitely argue there's some advantage to keeping a lower insurance limit, although it gets murkier when we bring behavioral considerations and "too big to fail" into the picture.