I would modify what you've said, software can be open source and be totally laden with patents, but free software (especially GPLv3) probably can't.
Continuing my argument: Claiming Mozilla are poor as the GP did is a bad argument. Claiming that the software can't be re-distributed is a slightly less bad argument, but only for larger distributions.
The pool sets certain payment stages. If your distribution of the codec has fewer than 100,000 users there is no license fee payable at all.
Also ,the license fee for h.264 is capped at $6.5 million, so Firefox can distribute it to anyone and any smaller project (less than 100,000 users) can distribute it as well.
It's the mid-sized redistributions that can have trouble, because they will lack Mozilla's money but have larger user bases that might trigger the fees.
If they just ship Mozilla's official binaries the license would be carried along. The only problem is that distros insist on recompiling stuff that already works fine.