The Enclave is pretty much done for, that is if you considered them the continuity of the State to begin with.

Vault-Tek's plans with secret vaults and their own agenda seem to have died out as well.

Is there anyone left with the legitimacy to reestablish the country with some sense of continuity from the Pre-War time?

Short answer: no.

The Enclave as far as we know is the only faction to have any sort of possible claim to be the continuation of the U.S. Government. And some people dispute or disagree with their claim.

Either way you look at it though, the U.S. is dead and gone now.

Also wasn't the Enclave the ones behind the Vault experiments? Pretty sure they were.

They were.

Moving on, is there no hope that some members of government managed to survive? It was clear that facilities like the Hidden Valley Bunker were built for continuity of government

According to Eden, these sorts of facilities were evacuated to the Oil Rig after the war. Which is why there was no-one in Raven Rock other such facilities:

"In the decades following the war, I watched as the remnants of the Government retreated to the West Coast."

Eden was specifically built to serve as the AI that controlled and monitored the relay array for "various installations around the country." So its safe to assume he observed these bunkers being emptied out when the orders came to rally on the Oil Rig.

Anyone who didn't retreat west either died out or deserted and left on their own long ago.

Hmm.This is interesting,at the very least it proves that the US Govt was by definition entirely into the Enclave, since all remnants eventually moved to the rig?

I just have a hard time thinking they'd bet it all into this plan A,the Enclave. There has to be a secondary organization descended from the government and otherwise organizations descended from it,sort of like the BoS came out of the US Army.

The NCR.

The NCR is not America. They don't claim to be what is left of America. The NCR is mearly emulating the American system of governance to a point. The NCR was not founded by members of the United States government. It was founded by people from Vault 15.

The NCR is more like a new country based around the US .

Why? Why spread valuable resources around when they need to be concentrated and consolidated in order to survive in the post-apocalypse.

They didn't put all their eggs on the Oil Rig during the war. Various installations were utilized during the war. However, given the limited communication and even more limited resources, at some point you have to rally together in order to get anything done. It makes perfect sense.

We can speculate all we want, but that's all it is: speculation. As far as we know, no such group exists.

The Institute is the only other organization I've heard of with possible connections to the pre-war Government. But they were a pre-war University: not a bunker for politicians and government officials. And no doubt they have no interest in the United States as a nation. Claiming the Institute to be the heirs of the U.S. is sort of like claiming House is a remnant of the Government because his corporation (RobCo) contracted out to the U.S. Military.

The NCR is its own sovereign nation. It has no connection with the United States except that it exists inside its former territory.