Yes. Page was a friend of the Secretary of War, Henry L. Stimson, and was called up by him when the first person assigned to write the Presidential statement (William L. Laurence of the New York Times) was found inadequate (Laurence did other work for the Manhattan Project, but his prose was tooA florid to pretend to be the President).

Page worked on several drafts, which were given to the Interim Committee (chaired by Stimson) and to the head of the Public Relations division of the Manhattan Project (William Consodine), both of which made suggestions. Page redrafted the statement several times along with General Leslie Groves.

A lot of committee work and thought went into the first statement. It was meant to do a lot with a short amount of space — convince the Japanese that the atomic bomb existed and was real, call on them to surrender, inform the American people about the atomic bomb, and lay out the foundation for a postwar effort to control the atomic bomb. It was part of a larger program of "Publicity" which included the Smyth Report, a major statement by the Secretary of War (which goes into more detail than the Truman statement does), and pre-written newspaper articles by Laurence that all newspapers could use as-is or as the basis for their own articles. It was a very carefully-coordinated information blitz, run from the very top (Groves and Stimson). Shortly after the order to use the atomic bomb was made, the final draft was shown to Truman, who approved it personally with no changes.

As for validating it: this is all in chapter 3 of my forthcoming book on nuclear secrecy in the United States! :-) The relevant documentation is in the microfilm collection known as the Bush-Conant files. The only secondary source I can think of off-hand that goes into this sort of thing is Alperovitz's The Decision to Use the Atomic Bomb because, despite the faults of his book, he is one of the only other scholars who has spent a lot of time on the question of how the Manhattan Project officials managed the official narrative.

(As an aside, Page only wrote the Hiroshima press release. The second speech about the bomb, which happened after Nagasaki, was drafted initially by Truman himself, although it was edited by his White House speechwriters. I have discussed it in depth here.)