I am a person who believes that conditional proposition. I am not sure if the theory of everything that describes the world is a deterministic one or not, so it is an open question. But are there any philosophers who believe this, and if so, who was the first in recorded literature?
It depends on the kind of possibilities you're talking about. If you're talking about logical possibilities this is clearly not true. There are conceivable states of affairs that did not occur.
I guess you're talking about possibilities that are not merely "in the head". If you're talking about nomological possibilities (what is possible given laws of nature) this is still not true if you assume "the initial state of the universe could have been different"is true. There are counterfactual possibilities corresponding to different possible universes with different initial conditions.
Now you could assume necessity of the past. Then your statement is true, by definition: if determinism is true, laws of nature determine the future state of the universe and so there's only one possible way the world could evolve.
Finally you could talk about metaphysical possibilities. Many authors since Kripke believe that they are distinct from nomological and logical necessities, in particular because laws of nature "could have been different"metaphysically speaking. In that case, your statement is false because there are metaphysical possibilities corresponding to alternative laws of nature even if the world follows strict laws of nature. But others think that metaphysical possibilities are the same as nomological possibilities (in particular, dispositional essentialists) and others that they are closer to logical or conceptual possibilities.
As to how long people have been thinking that, well, since there are different understanding of possibilities it's hard to say, but questions about necessity and determinism go back to the Greeks.
I'd say it's a common argument of many religious determinists that if there were an ultimate creative being, then every choice of that being would be the only possible one - of course, this is a very specific type of determinist.
From the scientific perspective - I guess the most cited example I've heard is "Laplace's Demon", named for the great French mathematician Laplace - he argued that if a demon (it would probably be easier for us to think of it as a computer) knew the position and velocity of all existing particles in the universe, it could then use the data together with the findings of Classical Mechanics to decide exactly what the future must be. I think that this is largely the argument you present, although perhaps not stated in the same terms.
From my layperson's understanding of modern physics, we currently don't have a known theory of everything, and some of our most critical theories are non-deterministic. However, there is one common possible deterministic physical model I happen to think is fascinating, and I think should be fascinating to anyone interested in determinism: Superdeterminism. Here is Wikipedia on the subject: