The OP while well intentioned is not bloody good enough, sorry.
It's all very well telling them to fire Roy; I'm sure they'd bloody love to - they're not daft, they said in the LFCTV interview weeks ago that results were unaccpetable & they've got worse since then. They will know we are restless.
FSG/NESV are bright men, wealthy men, used to making important decisions. However, in the past when faced with an important decision they've had a knowledge of the environment they're operating in; in this case though they don't have that knowledge (& probably have doubts about who they should listen to) & because of this they can't make a plan on how to make things better than they are at the moment.
Accordingly the main problem, as I see it, is that they still have no fucking clue about the game, they admitted as much themselves when they took over & they said they'd learn as fast as they could. However, they didn't foresee the situation where RH's position became untenable so quickly & they haven't learned about the game quickly enough to give them the networks & knowledge of how to retrieve this situation.
I remember, years ago a bloke I worked for asked me to bring him solutions not problems. It's a cheap line in many ways but in this case it's 100% appropriate; they need our guidance & this is why the OP doesn't go far enough. We need to present them with a solution. Not a list of what we don't want (O'Neill, Allardyce etc...) but a proper list of deeply suitable candidates with proper reasons as to their suitability that goes much, much further than Rafa Benitez.
It would need to detail, win ratio's, a cost/benefit analysis of each managers transfer dealings by comparison with other people in the same league. We'd need to age profile them (illegal I know but we need someone with time infront of them), it's need all that and a whole pile more that I can't think of when I'm supposed to be working...
If anyone has got the time to detail a way to critically evaluate, in an non-emotional way, the characteristics & personality traits, the successes & demonstrations of those successes I'd be mega grateful.
As no doubt would our owners.