Roy, where are you?
I renewed my licence at Malaga Tráfico office in May this year. If I had accepted another floppy bit of pink paper, it would have been issued there and then. I opted for a plastic card instead, which has to come from Madrid by post. I was also issued with a temporary document, but received the card in three weeks.
Maybe your local Tráfico office has messed up the application. Maybe Madrid have messed up. Maybe the Post Office have lost it. Or maybe the company you paid messed up. Wherever the mess has occurred, in my experience the only way to be sure things go right here, is to do them yourself, daunting and time-consuming as it may at first appear.
If at all possible, personally, I would go to the relevant Tráfico office myself, with the temporary document, and find out what's up.
Then, when you've sorted it out, go back to the company you used and demand your money back. Just out of interest, how much did they charge you for this non-service? Off the top of my head, I recall the renewal fee for the licence is about €17. My visit to Tráfico took up about three hours of my time tops, and the parking was free!