That's a management function.
If I took a Staff Engineer role and my job was to compensate for organizational dysfunction and fill voids left by underperforming managers, I'd be leaving that role very quickly. Either that or requesting a formal title change (or promotion) into an actual management position.
Hiring a Staff Engineer when you really need better managers isn't a good solution, especially because a Staff Engineer wouldn't have the necessary org structure authority to be an effective manager-of-managers.