My opinion backed up by my purchases, is that I really like the weight of the DA* 60-250 ƒ4. You still need a long lens for wildlife, but, to say you can't shoot wildlife with it is a bit of a stretch. Add the 1.4 TC to it and the focus breathing issue is somewhat alleviated.
Wildlife I've taken images of with the 60-250.
A bear, foxes, wolves, small birds, big birds.
It's not 600mm, but, niether is a 70-200. The one advantage to the 70-200 2.8s is, you can put a 2x TC on them. If you're not planning to do that the 60-250 will do nicely.
Not good enough or wildlife?
Or small birds
Well OK then, whatever. You still need something over 400mm from time to time, but the 60-250 is great way to compliment that.