Here's the quote: The man's position in the Executive Branch was extremely sensitive. It doesn't say he was working day to day in the White House or whatever. It's meant to convey that he had serious connections when it came to what was going on in the White House, that he was very much in the loop. Felt had White House liaison responsibility after the retirement of Deke DeLoach. That's a very sensitive position to be in within the Executive Branch.
Just google "mark felt" and "deep throat" you'll see all kinds of support for his being DT.
James Mann had private conversations with Woodward back then and was able to figure out he was positively an FBI guy (regardless of whether it was Felt or someone else).
Jack Limpert, who wrote the story back then that I excerpted for you, revealed later on that the source for the pieces was Frank Waldrop, who was "absolutely wired to the FBI," according to Limpert and, he was the former editor of the Washington Times-Herald, which was folded into the Washington Post in 1954-- which gives him a very good reason to piss on the Post's golden goose by telling Limpert way back then that he thought Felt was the man.
An FBI liasson to the Executive Branch would hardly be in a position to even know about Nixon's secret White House recording system, much less know about the gap on the tapes.
You might pause to consider that Justice Department prosecutors (who happened to serve as top tier Republican fundraisers) were likewise wired to the FBI.