I'll make it quick.
We registered at the Contrôle des Habitants on August 9. Paid a lot of money.
We went to the "Centre de biométrie et des documents d'identité" on August 17. We had our photos and fingerprints taken. Paid a little bit of money.
They told us we would get our cards in the mail in 2 or 3 weeks.
It's been almost 7 weeks now.
What's going on?

I have heard of such cases, very rare though - and for a reason. You should visit the canton office and check with them directly.