Can someone explain how elements are generally modeled to have formed from the big bang? And is there anything that it Jacob may be missing in the current literature?

Yep. Jacob's quite right about nucleosynthesis in the Big Bang, but that's not even close to the only nucleosynthesis pathway out there.

First-generation stars (called "Population III", confusingly) are thought to have contained almost no elements heavier than helium, which may have allowed them to stably reach much higher masses than the current generation can manage. Mid-weight elements up to the mass of iron are formed through one of the several fantastically complicated fusion processes which occur as older stars deplete their fusible hydrogen and start accumulating helium in their cores; carbon in particular is generated mainly through the triple-alpha process. Elements heavier than iron don't release energy when fused, so stars can't produce them in quantity; they're instead produced mainly by fusion events during the early stages of a supernova.