I wanted to give a fair amount of thought to your idea before posting a response. After all, if ... a6 is often a reasonable move in the Slav, then surely ... Qb6 is worth a look. Your analysis does seem to confirm that 6. Bxf6 does not work out well for White.
The problem arises after the other reasonable move 6. Qc2. After your suggested 6... Nbd7, if White plays 7. e3 we have transposed from a Semi-Slav into a Queen's Gambit Declined. Consider the moves played out in the following order: 1. d4 d5 2. c4 e6 3. Nc3 Nf6 4. Bg5 Nbd7 5. e3 c6 6. Nf3. If Black now plays 6... Qa5, this is the Cambridge Springs variation and it has a very respectable reputation. If instead 6... Qb6 7. Qc2 and we have arrived at the same position as in your line. In the Cambridge Springs, Black has a fair amount of counterplay which is unfortunately missing here. So, I think with best play, it leads to a somewhat inferior (for Black) version of the QGD.