The 70-200 f4L is a cracking lens for the price, it's very sharp, very light and it's fast to focus but it's just not really long enough for birding. It's okay for some wildlife shots like squirrels etc in a park but that's about it. This was the first 70-200 I had and eventually worked up to a 70-200 f2.8L IS MkII.
With the 70-200 f2.8L IS MkII it takes a 2x MkIII TC very well to give me 400 and that's the minimum I'd want for wildlife and birding. I use my Sigma 150-500 OS for 90% of wildlife and birding shots and I find it excellent.
I had a Canon 100-400 at the same time as the 150-500 and after a lot of testing I decided to keep the Sigma as they were both equal for IQ, but the OS system of the Sigma was much better than the IS of the Canon and the Sigma seemed to focus better than the Canon in failing light.