Honestly even the Legion getting into the Dam during the Second Battle isn't going to help them much in winning. The NCR regular troopers there IN the Dam are going to chop down Legions of the Legion from concentrated fire that doesn't need more than 6 weeks basic to be effective. Yes the Legion would have a melee advantage for a short time before they're shot down by the rest of the squad. To make matters worse NCR radio is effectively coordinating all the damn squads in the Dam making the Legion job even harder coming from the intake towers. Then move onto the traps Oliver has in his office along with the VR there and Heavy NCR troopers and that's a lot of dead Legion. I'm not sure they'd be able to get through all that without the courier.
Moving past that on top of the Dam you have the Rangers killing off the Legion's best fighters; Centurions. And laughably the Centurions are trying to match the Rangers in medium range fighting with their Anti Material Rifles. Considering the vast amount of skill the Rangers have in well [censored] shooting vs the Centurions I doubt the Centurions would make much ground. Camp Golf would also fall for the same reason as Camp Forlorn; the failure of NCR troopers. In this case the ending says they desert but Hanlon and the Rangers obviously survive and do quite well in holding off the Legion themselves. In any case I'd expect a strategic retreat in this case as well with minimal NCR causalities and probably more deaths from NCR court martial down the line.
The only NCR position that would be weakened/overrun would be McCarran and yes Oliver's strategy was counting on reinforcements from there but that wasn't going to happen anyway since the Legion blocked the road west. In any case the reinforcements aren't even needed. I'm doubtful without the courier the Legion could realistically kill ALL the NCR at Hoover Dam, especially since without the courier Oliver wouldn't call a retreat. Further the game endings don't really indicate they destroy Forlorn Hope or Golf. Without House/the Courier [censored] things up at that meeting at Legate's Camp the NCR wouldn't of mass withdrew from the region. And without the courier helping Caesar they sure as hell wouldn't be slaughtered in the same way.
Hell for example, Boone's unit; First Recon survives Forlorn Hope under the House ending because it states he thinks about rejoining it. So ending data and common sense tell us the NCR shouldn't be counted out when it comes to the Second Battle. The fact remains the majority if not super majority of NCR troops and Rangers would still be alive even after they lose positions like Golf and Forlorn. Further NCR communications and organization ensure disorganization won't follow. The only NCR position basically destroyed is McCarran but that's irrelevant to the Second Battle in the end.
The Legion wouldn't be able to hold Hoover Dam and certainly wouldn't be able to take Vegas. In lue of an order to retreat which only the courier can produce, the NCR would either win the Second Battle (which they have a high chance of doing really with a proper counteroffensive or even a strong enough defense during the Second Battle.) or die to the last man in the sands of the Mojave effectively destroying the Legion in the process.