I mean... it's fairly obvious in my eyes.
Up until Sunday at least the team was showing character and determination despite its obvious frailties. For example despite being very disjointed and playing poorly for periods, the team managed late wins against Villareal and Atalanta as recently as last week. The players might not know what they are doing under Ole, but they do seem to want to fight for him or the shirt at least.
Obviously that will only carry you so far. When you play against an expertly coached, cohesive, in-form, Liverpool team... then even with all the determination and good will in the world, you're still quite likely to get the pummelling you deserve for not being a team.
Now if Ole leaves and the current coaches stay under a caretaker manager, the team will not magically become more cohesive and less disjointed. That will take time either way. The only thing that might change in the short term is mood and willingness to fight. And seen as that wasn't the problem to begin with, I think that they could indeed start to lose interest and not bother until a proper manager comes in. Why would they fight for Carrick or Phelan more than they did for Ole, when they know it's only a matter of time before the caretaker is out on their arse anyway. And then we'd be in worse free-fall losing not only they games we don't expect on current form to win (Liverpool) but also the ones we do (CL group games against Atalanta and Young Boys). So yeah, it could definitely get worse imo.
You're pre-supposing that same spirit and attitude is still in them after the biggest humiliation some of them have ever experienced in their entire professional careers. That I can say that solemnly should tell you how bad things are right now and what you're suggesting could compound that further and lead to a point break where the season is a complete write off, and a new manager is having to pick up the pieces back to even a neutral marker from a severe negative, just to get things underway.
It's a calamity, and keeping the man responsible for it in charge for further potential humiliation(s) makes no sense on any level given if the worst comes to the worst, it's not even a case of told you so's over gross, gross negligence and incompetence.
New men in charge for an interim brings a modicum of new hope and belief, and at the very least, acts as a reset and clean slate from the doldrums of yet another repeat of a well-worn script that has reaped hardly and dividends for the season up to now.
To your bolded - Ole is the man in charge who ultimately makes the decisions that everyone else has to abide by; you're assuming those men you mention follow the same ideas and notions when in fact they may have their own vision and idea of what they want to do with the team, tactics, formations, personnel, in-game actions and so on and so forth. We also have no idea if they are as rigid and inflexible as Ole and persistent in trying the same things, repeatedly, in the face of them failing conclusively.
Normally, there is merit to the other side of such a discussion, and if desired, I could argue it from your POV or from the POV of it being beneficial for nothing to change, but I am certain any point in favour of Ole staying on I could absolute tear to shreds such is the disparity in what he is giving as opposed to what is needed. We're under-performing by such margins it is indefensible by now. We are not good at anything, anything at all, to even clutch onto and have as a hill to die on for, or in favour of, Ole.
Like I said previously, really examining what's gone on is dragging him over hot coals for no good reason by now. I'd like to point to there being benefit to him staying, even in the interim, but there aren't any - the players have lost faith in him, and our football has not a single redeeming factor to it. Not one. Do you know how rare that is? The worst kind of bingo full house you can get in football.