^ What it says in the title. Take the Courier out of the equation, what happens to the various characters and factions we encounter in the game?

For instance,
- the Powder Gangers overrun Goodsprings
- the McCarran monorail gets bombed

...etcetera. What happens to Benny and his plot, for example? What happens to the Mojave?

Without the Courier House will have to find someone else to enter the Tops and get the chip or enter with his securitrons, either way, Benny flees to The Fort at which point he gets captured and House's plans are foiled.
No one will be allowed into the Lucky 38 as House will likely shut the gates down, meaning no one can assassinate him without blowing the whole thing up. (Speculation, but why wouldn't he close it down if his plans just got screwed over, why would he open the doors? He only opened them for a reason and that reason is dead.)
The president is assassinated, the Misfits are screwed, the Great Khans help out Legion, the Omerta kill the strip, the White Glove Society stays out of it, the tension with NCR and Kings reaches it's climix.

Lots of people get hurt.

Goodsprings falls. Or not, the Powder Gangers weren't super tough.
Primm remains in chaos.
Novac gets overrun by the Legion (and/or ghouls).
Monorail gets bombed.
Forlorn Hope remains Forlorn.
Benny is probably caught by the Legion.
House is stuck, can't do much.
The Omertas attack the Strip.

Basically, Legion wins.
Tbh, I'd even be willing to bet Caesar survives his tumor. The Legion is more than willing to throw lives away to save Caesar, and they know where to find an autodoc. They could probably even convince the White Gloves to join them. Legion really doesn't seem to NEED the Courier...

Id say goodsprings survives, they are pretty stout hearted. In the end though I'd give the legion the nod due to the contributions that the courier brought to the ncr.

Yeah Goodsprings could potentially survive.
It depends on when Legion attacks and when PG's have had enough of Goodspring hiding Ringo.
If Legion attacks shortly then they'd probably scare away the PG's.
If they take their time or they have to clean up other things, like NCR and Mojave Chapter for example, then PG's might not leave immediately and will attack Goodsprings.
Has anyone done Ghost Town Firefight and only recruited Sunny and Ringo? I think they're pretty outnumbered if you don't get involved.

As to the Auto-Doc, they know the location for the device IIRC, Lucius tells you right?
So they'd most likely send a group to V34 and get it, or use ghouls to get it, promising them asylum if they help Caesar.

And yeah, Legion does not need the Courier really.
Boomers?
Caesar considers that a bonus, he might send a attack force to their base but after them getting wiped out he'd leave them alone, unline Lanius.

Mojave Chapter?
Caesar states they have taken out chapters before and can do it again, it just takes a lot of legion lives to do so and would rather have it done with precision.

White Glove Society?
I think they'd let them stay as citizens as long as they don't openly oppose Legion, and might even integrate them after the battle is done.
Doubt they'd kill them off right away though.

Really though, NCR has nothing in their favor.
Nothing good happens to them if the Courier is taken out of the equation.
They don't ally with MC, they don't ally the Boomers and can't get in, they might try to take out the Great Khans but the GK has the upper hand in their defense, they might try to foil Omerta's plans but it's debatable.

And in terms of lore a legionnaire is far better trained than a trooper, they have far higher discipline than a trooper and they are zealots to the son of mars where as in NCR we can see people like that junkie in Forlorn Hope and the Misfits at Golf.

So Legion wins.

NCR "could" try to pull a Hanlon.
But it ain't Graham behind the arm of Legion this time, it's Lanius, and they ain't gonna fall for the same trick again.

Benny's plot still fails.
House employs someone else.
The Legion continue their advancment across the river and across the Dam.
Legion succeed in forcing out all NCR occupants from New Vegas.
NCR fail to get a grip of their soldiers.
Halons plot is never uncovered.
McCarran keeps getting attacked relentlessly by the Fiends until it's overrun.
Fiend leaders are never killed.
The Khans are assilimated into the Legion and happy with their revenge on NCR.

etc etc

Yeah, Chet would most likely survive... but with the exception of Sunny Smiles the Goodspringers don't seem particularly keen to fight. I see them just accommodating Joe Cobb and his buddies as much as possible without the Courier.

I agree. Without the Courier to tip the balance, the stalemate in Goodsprings remains intact. Neither side gets the extra help to overrun the other, so the status quo remains.

In fact, this probably applies to the entire battle for Hoover Dam. It's the Courier's intervention that tips the scales one way or the other, so without that, the stalemate will probably last for some time.