Another vote for "it depends on where you sit". A couple years ago I sat further back and had the nifty fifty on and found that not to be ideal for me.... it was too wide. This year I put the 70-200 f2.8 on from the start but immediately changed to the 17-50 f2.8 because I was sitting in the front.
I have a 70-300 F3.5-5.6 that I plan to use for animal shots since they are usually well lit.
Uhm.... I used that lens on the safari and the trails in the past and heck was I glad I used the 70-200 f.28 this time around. Some areas may be well lit BUT... yes, there had to be a but

First, while the safari is well lit in some areas, the safari truck bounces you around so much that even in bright light, you will need a very fast shutter that I was never really able to achieve with the 70-300 unless I upped the ISO quite high or the truck stopped (which as I have encountered on our last trip and 7 rides on the safari, they don't tend to do as often anymore as they used to). Often, the animals will hide in the shade as well, so again, large aperture and fast shutter to get that. Secondly, some of the animal trails are still very dark since there is quite a lot of vegetation (for example the gorilla area). They have two hyperactive baby gorillas at the moment that if you want to catch them, you will need a fast shutter in that poorly lit area. Even my f2.8 with ISO 1600 was cutting it really close at times with those two. If I were you, I'd just leave that lens behind. Will you miss the extra reach? I am quite honest with you. I was really, really fretting over losing the extra 100mm since I had gotten the 70-300 just for that purpose. I just knew I was gonna miss it, even before I had made the switch. But once I had seen what the 70-200 can do and that when I crop those pics, they still are better than most of the ones I had taken @300mm with the other lens.... I haven't touched the 70-300mm since. Just something to think about. I've also used the 70-200 on shows like Frozen, Nemo, Beauty and the Beast, Indy, Halls of President, etc. Again something I wouldn't have been able to do with the 70-300.