Hmmm personally I would be a bit skeptical. If you ask me I would bet that is the motor making the noise and perhaps by adding a bit of tension (you can see that the original was loose) this has temporarily resolved the issue. It is up to you really. See how it runs without the belt if it is relatively quiet then I guess it could be left.
I personally strip all motors now if nothing else it might just extend their life and definately makes them run sweater, however it is a fidely job, you have to watch the brushes and that tiny circlip and the washers!
Belt kits can be sourced from Dillen (Martin).
Just out of interest, when you spin the shaft with your fingers (no belt attached) how long does the motor spin for? Should be a good few seconds!