Yeah but in the case of the hit squads, they were sent after you. So when they failed to report back it is safe to assume that they were killed, by you.
Also, by the time you enter LR you are pretty much well known. If anyone was capable of launching nukes to destroy a nation, it would be the courier.
Again, how does Caesar know for sure it was me and not NCR Rangers and Powder Gangers in the middle of a battle and they got caught in between?
But yes, I agree that's how they must "know". But isolated, dead Powder Gangers aint tellin no tales. That's real talk.