I've used the 300/4 IS for many years and many, many shots. Probably upwards of 150,000. It practically lived on one of my 7Ds that I shot with for over five years (about 150,000 clicks on each of my two 7D).... and before that on my 50Ds... occasionally on my 5DII over the years.... and most recently on the 7DIIs I'm using now.
I originally bought it as a backup for my 300/2.8 IS, but found that I use the f4 lens far more often because it's handholdable all day long and leaves me mobile, where the 300/2.8 can only be handheld for short periods of times. I considered the 400/5.6L instead, it's a great lens too... but the 300/4 has IS and the 400/5.6 doesn't.
Plus I use the EF 1.4X II with the 300/4 at times, too... so have two useful focal lengths with it (i.e. 300/4 and 420/5.6 when the TC is attached). The lens works very well with that teleconverter...
The 300/4L is very fast focusing and more than capable of keeping up with moving subjects....
With distant, rapidly and erratically moving subjects (such as the hawk below), your biggest challenge will probably be keeping it on target.... no fault of the lens!
I bought mine lightly used about 7 or 8 years ago and have since used the lens quite hard myself. In fact my 300/4 now needs service... Not sure if it's the IS or the AF, but something is a little wonky when watching the image in the viewfinder... seems to jump around. It still makes sharp, in focus shots, though. It's just pretty difficult to keep right on target. I'm not really surprised if it needs repair, though, since it's certainly been used a lot! Probably only my 70-200mm see more use (one of those is often on my other camera).
The 300/4 is not as sharp as the 300/2.8... but hey, it only costs about 1/4 as much! And it's pretty darned good... I figure if I can count eyelashes on a subject photographed 100 to 150 feet away, a lens is plenty sharp for almost any use.
The 300/4 IS was one of the very first lens models that Canon added stabilization to and it still uses one of the more primitive forms of IS, which technically should be turned off if the lens is locked down on a tripod. However, I never use it on a tripod. I either handhold it or use it on a monopod, at most. So I really can't recall ever turning off the IS. It's probably good for 2 to 3-stops worth of assistance.
I have recently been using the EF 100-400 Mark II. And, yes, the zoom is more versatile. However, it is about $600 more expensive, not quite as fast focusing as the 300/4L IS. And it's about 1 lb heavier, as well as larger. My arms and shoulders are a lot sorer at the end of an eight hour shooting day! Plus the zoom is f5.0 or f5.6 at 300mm and longer focal lengths (f4.5 at best, at 100mm). Great lens, too, though!
The 300/4L IS was the closest focusing of all Canon lenses longer than 200mm... until the 100-400L II was introduced. That now gives higher magnification, close to 1/3 life size (but at a longer focal length). The 300mm is slightly less, but much closer than the 300/2.8 and excellent as a "long macro" (okay, make that "near macro"). This is slightly cropped, but was shot with the 300/4 pretty close to it's minimum focus distance...
A couple
minor nuances... First, in certain situations the 300/4L IS renders a slightly "busy" background.... the 300/2.8's "bokeh" is nicer. With the 300/4 it shows up at certain distances and f-stops, most noticeable when there are scattered highlights in the out of focus areas. (Actually, the 100-400L II does the same thing.... as do some other lenses.) It is actually pretty easy to deal with in post-processing. For example, check out the slightly "coarse" appearance of the cars in the background of this shot (uncorrected)...
Another thing, the 300/4L tends to render specular highlights with a magenta color cast. Again, it's no big deal and if a problem can be dealt with easily in post-processing. You can see it in this image, in the catchlight in the bird's eye (see the detail from the image)....
Obviously, I really like the 300/4L IS... I'm not about to give mine up, even though I got the 100-400L II earlier this year.