Is this the kind of rumor that starts a "run on the bank"?
I imagine any such event would require a bank to massively divest which could trigger a market event as well
Lender of last resort makes runs less likely and lossy. It doesn’t prevent them altogether.
Managed buyouts are preferred partially so that the government does not have to deal with the logistics of running a retail bank, but mostly because if a bank with sufficient funds takes over then there is no interruption to depositors’ access to funds.
Buyouts offers that cover all insured funds are typically cheaper than directly paying out. Indeed, most commonly the buying bank will offer to take all the full balance of any insurable account, not just up to the insurance limit. The logic here seems to be that the extra from taking on those balances is usually pretty small, and it does save work on the FDIC's side as there would be fewer creditors. Thus it makes bids with that include full balances more attractive than those that don't, increasing the chance that the buyer's bid will be selected for final resolution.
Of course, occasionally the winning bid is limited to just insured balances, and very occasionally, the cheapest option is for the FDIC to directly cut checks to handle the insurance side, and proceed as receiver in selling off any saleable assets and paying of creditors.