William D Middleton provided the canonical account of the Waldorf platform in his book Grand Central: the world's greatest railway terminal (San Marino CA: Golden West Books, 1977), as follows:
An article titled "Discovering the secrets of Grand Central Terminal," from the Journal-News of 9 Sep 2001, describes the freight elevator used by Pershing.
Its location above the tracks permitted the celebrated hostelry the unique distinction of its own railroad side track in the basement, so to speak. Officially identified as Track 61 in one of Grand Central's storage areas, with a freight elevator providing access to the hotel, the siding was used on occasion for the arrival or departure of distinguished guests traveling by private railroad cars.
General John J Pershing was the first to use it, on a visit to the city in 1938. During the 1944 campaign Franklin D Roosevelt gave a foreign policy address at the Waldorf and then descended into the "basement" to the presidential rail car for the journey home to Hyde Park. On other occasions the siding has been used for such diverse affairs as a 1947 "debut at the Waldorf" for a new 6,000 horsepower diesel locomotive, or for a 1965 "underground party" for pop artist Andy Warhol.
[Metro North spokesman Dan Brucker said that President Roosevelt's] "armor-plated Pierce Arrow car would drive off the train, onto this platform and into the elevator, and it would bring him and his car into the hotel garage." ... The 6-foot-wide elevator, built to accommodate a 6,000-pound armored car, is kept in shape by elevator mechanic Darick Jones. Once at street level, Jones yanks the elevator gates open to reveal 49th Street. Driving an automobile with a slim profile, one could still make a sharp, right U-turn into the Waldorf garage. . . . The locked entrance to the secret station is down a stairway concealed behind a brass door marked 101-121 49th St, below a sign that reads "Metro-North Fire Exit".
The Roosevelt story has taken on new dimensions here, with the automobile riding the train and taking the elevator. Recall that Middleton mentions just one use of the elevator. Many re-tellings of the tale now assume that Roosevelt used the platform routinely. For example, take this account in an academic book, Grand Central Terminal: railroads, engineering, and architecture in New York City by Kurt C Schlichting (Baltimore: Johns Hopkins University Press, 2001):
When President Franklin Roosevelt stayed at the Waldorf, his train would stop on the upper level of the underground train yard directly under the hotel. This enabled the president's aides to carry the paralyzed Roosevelt through a special door and then by elevator directly to his room, avoiding the public altogether.
Unfortunately I have been unable to confirm even the one use that Middleton refers to. It must be Roosevelt's visit to the city on 21 October 1944 to address the Foreign Policy Association (see the New York Times). He made a very public arrival with a four hour tour of four boroughs in an open top automobile in the rain, to let the people see him. The secret platform on that trip was outdoors at Bush Terminal: Robert Sherwood, an eye witness, says that's where the train arrived, in his book Roosevelt and Hopkins. Nothing is said about a carfloat, so the private car must have gone through Penn Station and then via Long Island lines.
The Times does not disclose Roosevelt's hotel, but a followup story puts him still in the city the next day, attending a dinner party in his honor at the Astor Hotel. Did he then go to Hyde Park? Roosevelt made no public appearances between the dinner in New York on 22 October and an afternoon press conference in Washington DC on 24 October, but that still doesn't give him much time, and his health was weakening, despite the impression he gave the public. He died in April 1945 of heart disease. It seems most likely to me that he journeyed to Washington on 23 October, ruling out use of the Grand Central platform. There is perhaps still room (but not much) to argue.
I found a BBC interview from 2009 that talks about FDR using it, car and all, and goes down into the bowels of the station with that mentioned Brucker. As a note, I didn't take a tour of Grand Central, although some are offered - has anyone? What do the tour guides say?