Stepping back and looking at Sitting Bull's advantages:
- Dog soldier: As you pointed out, they are amazing for defensive purposes, make you fairly safe from barbs as soon as you reach Bronze Working, vs. having to connect a copper as well. The existence of this UU kind of devalues his PRO trait and Totem Pole though as he wants Archery in even less cases than the average Civ.
- Starting techs: Agri is almost always good. Fishing is maybe something a PHI leader would rather *not* have on this map, since not researching fishing opens up the opportunity for a great scientist to bulb Engineering. At least, Fishing saves you something like 20 beakers when you get to research Pottery.
- PHI: More great people means you can rush one of the later units (such as Cuirassiers) faster with a comparatively smaller empire. Having an abundance of great scientists encourages you to get an academy in your capital (especially one like this) where other leaders have to save it for bulbs. Taking advantage of this trait requires mostly food and happiness.
So PHI is better for the long game, which doesn't mean you'd never attack early, but here I would also consider that you have a rather friendly bunch of opponents, with the notable exception of Shaka of course. None of the others can start plotting against you when pleased (
https://forums.civfanatics.com/threads/civ-illustrated-1-know-your-enemy.478563/), and you can get them to pleased by giving them techs (or trading at a large advantage in their favor).
The other thing that synergizes very well with PHI here is Pyramids. Probably needs to be built before T80 on immortal and you have potentially strong competition for it (Ramesses).
Now, I'm not sure that's a good plan, but let's say you settle between gold and stone (forever size 1 working gold), then city 3 on the jungle W of cow. Cow city gets a potential 7 hammers per turn (plains hill mine + cow + city center) at size 2 and 8 forests (saving 2 from the capital). Then it can build pyramids in about 13 turns with 2 full-time workers. If you successfully get the pyramids the culture grabs the corn and bananas. The tech path in this scenario is probably something like Wheel - AH - Pottery - Masonry.
EDIT: Now I think it's a bad plan, because the eastern stone is on desert. It would take 8 worker turns for a quarry then 3 for a road. You could still put the 2nd city on jungle near gold like
@Fippy suggested, 3rd city on jungle W of cow and use a library to expand borders to corn and bananas, and 4th on coast SW of sheep. Settling gold city first in addition to the advantages pointed out by
@Fippy, gives you more time to research AH, possibly fitting in the Wheel before it, which really helps commerce.