That’s nice to have a new topic to debate in 
But when saying “debate”…
Speaking about the start menu, I’m not sure this would be such an interesting feature. I mean, if we begin with 10 members in our tribe, on a random map, we don’t know what resources would be the best. What if you choose good hunters but end up along a shore, or the contrary? So, my best guess is that we should keep to randomness and avoid piling up modifiers.
On the society thing, however, I think choices should happen pretty fast. As you noted @Mishab there will necessarily be choices to be made very early, to define who should be the next family head for example. So as soon as your first head dies, the best way would be you have a choice between the oldest member, or the oldest male/female, or the firstborn son, etc.
Then, when peculiar occasions arise (for instance, no possible candidate because of a lack of son, etc.) you’d have to change your system.
Concerning a broader power structure or hierarchy (let’s call them “kings” or “chieftains”), we finally have the same issues as for hunting or fishing modifers: what’s the point of selecting before even launching the game how your tribe will be ruled, if for hours you only try to survive with your 10 members? A “king”/“chieftain” only comes in existence if you’ve got a number of families, and a number of family heads, not when you have 10 or 15 citizens.
But on the principle, I agree with you that having such choices triggered would be totally nice.
Another thing though: all what I said above is for the first release of the game. If we ever have (I hope so!) DLCs released later, where you could play e.g. in Egypt or in Greece without needing to go through the Neolithic era, it could be interesting to choose a culture or even create a random one (e.g. take the power-monger Assyrians, or spiritual Egyptians, or technical Greeks, or lore-lovers Babylonians, or barbaric Cymmerians, etc.).
One last thing: @kelstr
I’m not sure what you mean when writing “tribal color”: is it skin color, or a symbol?
I hate those games when you end up with a symbol you don’t like, but for Neolithic that’d be probably anachronistic to have such emblems, save maybe for spirits or gods.
However, I admit that maybe such things as tattoos or body paints showing on each character their family, then maybe later their tribe (when there are several families, later in game) would be a damn cool thing!