That's an absurd suggestion. Mozilla can't just "pretend" that patents and licensing fees don't exist. FFMPEG gets away with it because they're not a commercial entity. Do you really think Google would have willfully paid to license H264 if they could 'just use ffmpeg' instead?
Until W7, Microsoft didn't ship a decoder for H.264. Neither Microsoft nor Apple ship support for MPEG2 (you have to pay extra), and nobody supports Xvid.
Yet Microsoft's Windows Media Player 'just uses ffmpeg' to play these formats when you have one of the many DirectShow packages installed, and has since the beginning. Apple's Quicktime does the same when you have Perian installed. Opera uses GStreamer for <video>, and does the same when you have gst-ffmpeg installed.
Google is mostly paying for H.264 decoder licenses because they were already going to have to pay broadcaster-level encoder licenses.
Opera only uses the system gstreamer on Linux, on Windows (and probably Mac when they get around to it) they bundle their own gstreamer that only supports Theora.
I can see Firefox doing similar on Linux, so if it's just (ironically) the Linux geeks that are desperate for the patented codecs then I wouldn't worry too much about it. For the 60% of Firefox users on XP though, Mozilla aren't going to rely on system codecs that aren't there.
I'd be surprised if you couldn't drop your own gstreamer plugins into Opera's directory on Windows/Mac.
You'd be surprised at the hundreds of millions of XP users with codec packs installed already. For the remainder Mozilla would just need a nice 'codec not found, go here' dialog like they already have for Flash.
You're completely ignoring the licensing part of the problem. Google can use any library they want to decode H264 because they paid for a license. The problem is not the library, it is the license.