Per some insurance maps the station was torn down by 1917 except for the outbuilding which became the little gas station. If the Bostonian Society's search wasn't down, there are dated demolition photos of the station's train shed that would give the exact date.
The plans for the station are on file, with an appointment, with most of Peabody & Stearn's work at the special collections desk of fine arts in the BPL.
The Hotel Statler officially opened in 1927 as the first hotel in the world to offer radio service in its guest rooms. The back of the theater in those photos has a billboard for the future home of the hotel, so whenever that photo is dated is likely the demolition year of the theater.
The NYT has this abstract from way back when,
"A NEW THEATRE FOR BOSTON. PLANS DRAWN FOR A BIG HOUSE ON PARK SQUARE. BOSTON. April 11.-The latest popular theatre project la by James Stevenson, who contemplates building on his Park Square property from plane submitted by C. H. Blaekwell, architect of the Bowdoin Square Theatre. The main entrance will bo on the corner of the triangle opposite the front part of the Providence Station, and fronting upon the square. The vestibule will be '25 feet wldo and free from all supports. A round tower is to flank the facade on each side, with an open arcade between, over the stores . This will be used as an open-air promenade. Inside the vestibule there will be a lobby of ample dimensions, and from the lobby will open a smoking room and series of arcades, beyond which la to be a garden on the land behind the stores Columbus Avenue. This uow unused land Mr. Stevenson proposes to lay out In a park, where one may stroll between the acts. Mr. Stevenson has also arranged for a hippodrome in the " Battle of Gettysburg " building on Tremont Street. It will be run much Ilka the old Barnnm hippodrome in New-York. There will also be a museum of curiosities in connection with it. i* ___________ rn ____________"
