They can probably afford to pay the license for themselves, but Firefox is "libre" software, licensed (among others) under the GPL. If Firefox requires an H264 license to be usable, it's not a real open source project any more.
Mozilla certainly have the ability, but I agree that for principle's sake they shouldn't license it.
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.
In fact, even if Mozilla did pay a non-royalty-free patent license to include the codecs in Firefox, that license would probably be incompatible with the GPL/LGPL, meaning it would no longer be possible to distribute Firefox under its current MPL/GPL/LGPL tri-license. (I don't know the MPL well enough to know whether Firefox would still be distributable under the MPL alone.)