Surely if all the factions are meeting at the same place for a fight, the Boomers would just stop a kilometer or two away with their artillery and blow everyone else to hell.
unfortunately you are imposing your own restrictions. Although the posited question was perhaps crudely stated it can not be inferred that a free-for-all was intended. Certainly it wouldn't happen in the natural course and is an artifice introduced to support your conclusion. In addition, just as an air force can win a battle but not a war, neither can the artillery. it takes foot soldiers to clear, take and hold territory and no war, including the Great War of 2077—in which nobody won, has ever or will ever be won without them.
Mr. House is at a serious disadvantage without the Courier. Since Benny had already neutered Mr. House by stealing the chip and conspiring with Yes-Man, the absence of the Courier presupposes he has a better chance than Mr. House to take control of the Securitrons, and ultimately the Mojave. However, since is captured by Caesar his chances have been seriously diminished.
The only serious flaw in the current story is why Caesar has not pursued destroying the Securitron Army himself. This has not been adequately examined in the story.
The most probable scenario is that the chip stays with Caesar, Mr. House and Yes-Man are successfully neutered, the true use of the chip dies with Benny. Mr House manages to repel the Legion and maintain the Strip for a time until Caesar destroys the dam and lays siege to the Strip. Without energy or resupply Mr. House is doomed.
Caesar and his Legion reign terror over the Central West until the BoS and the NCR from the West and the BoS and Enclave from the East squeeze him like a teenager's zit.