I studied this P&F last year.
My thoughts:
Firstly it did wonders for my control of trills. The prelude has them in both hands and they really do require a very fine balance. I also found the prelude an excellent exercise in piano-pianissimo playing. This prelude never struck me as being one that should sound forced, so it certainly helps with the development of a singing tone in the softer dynamics, so by extension it is also an exercise in touch.
In my opinion, the fugue is actually one of the more difficult ones from book 1. If you are using a Henle edition, the last system on the first page of the fugue was particularly difficult to complete with all four voices together, however that was only where i encountered the most difficulty.
I also like this piece as I feel it's a side of Bach we don't see too often. It reminds me very much of the prelude of the Bb Partita.
But to answer the initial question I would certainly study this work. If your teacher is less than happy about it, perhaps ask why? Or simply don't tell him/her that you are studying it.