Whoops, you're right, I was looking closely.  Your T is the equivalent to the operator for the 1d quantum harmonic oscillator, which means that its eigenfunctions are Hermite functions, which are the same as the eigenfunctions of the Fourier transform, and the eigenvalues are right too (up to a scale factor, which I haven't looked closely enough to check).  So yes, it does indeed look like this might give the FRFT.  This does seem like a nice way of doing it.
If it works like it appears to, I would be surprised if it weren't known, since the FRFT is often defined precisely by looking at Hermite functions, and the connection of Hermite functions to the harmonic-oscillator eigenproblem is well known.  However, we would have to dig up a clear reference before we could put it into the article.