He needs more than a majority of Tories, though.
Indeed, he needs almost all of the Tory MPs to support him (at least pay lip service). Only he can hardly be called a unifying character.
True, but at any point in time any Conservative leader will have a significant minority of the party against them.
2/3 actual support combined with the party whip for the remaining 1/3 may be sufficient to block any attempts by the opposition to delay, derail or even stop Brexit.