I own an EF 400mm F5.6L lens and and highly recommend it. IMHO, this is the one of the best (if not the best) affordable birding/wildlife lens out there in the market... good image quality, fast AF and affordable.
IMHO, AF of EF 70-200mm F2.8 II + 2x is very, very slow... a lot of small birds move very fast and you'll miss a lot of opportunity .
EF 300mm F4L IS is also excellent, but more expensive than EF 400mm F5.6L and you still need to add a 1.4x tele-extender which is not cheap and slow down the AF.
The only "disadvantage" of EF 400mm F5.6L is no IS, but not really an issue to me because I always use a tripod as I've problem hand holding a >= 300mm super telephoto lens even with IS and my current DSLR (60D) doesn't really perform well at high ISO (I seldom use above 1600).
Anyway, for your reference, the attached is an image of Collared Scops Owl I recently took with the 60D + EF 400mm F5.6L (Tv: 1/20s, Av: 6.3, IS 800) and a 100% crop over the focus area. Process with DPP using standard setting and brightness +0.33 and resize in PS. No additional sharpening had been done.
Happy shopping and have a nice day.