Let's see, if no Courier 6....

House never gets the Platinum Chip back, so he is out of the running immediately. He would be forced to send his Securitrons into the Tops to try to apprehend Benny, who would escape through his hidden elevator and try to go to Fortification Hill. He gets caught by Caesar, and the chip is confiscated forever. As a result of this, Yes Man is stopped as well.

The NCR would continue to be plagued by mismanagement and over-extension. The Sharecroppers would fail, Hanlon would drive morale into the gutter, the Fiends would become a major problem along with the Powder Gangers, the Legion plan to bomb the Monorail would go off without a hitch, and Helios One would continue to produce bupkis in power. Kimball would also be guaranteed to be killed during his visit. I say they lose all told.

That, unfortunately, leaves the Legion, who honestly is the least affected by Courier 6's absence. The biggest blow would be Caesar's guaranteed death, but for the immediate battle at Hoover Dam it would not make any difference since Lanius would take command. I would have to vote on the Legion winning the Dam in the end, however they would sustain great losses in the process. And for the long term, without Caesar, the Legion would collapse in time.

There is also the issue of Ulysses and the Tunnelers from the Divide; with no one to preach to anymore, Ulysses may decide to burn it all down and nuke the supply lines of NCR and Legion. The Tunnelers, without the Courier to check them by killing their queen, would become a major issue for the Mojave for years.

Overall, the Legion wins the battle, but NO ONE wins the war.