As always, thank you for pushing my thinking.
As with most things, I think this is complex.
You are correct in your understanding of my argument: People ought not simply make the purpose of their company into the purpose of their life; nor expect their company (or any other external force or entity) to give them purpose. Or, said differently, ideally people will identify and pursue a purpose independently of a job; and part of how they pursue it should be by choosing to work for a company that gives them the opportunity to pursue that purpose.
Note: When I say "pursue" here I do not mean pursuing an answer as to what one's purpose is. I mean once you know, you now need to make it happen and that can be construed as a pursuit.
Now onto more complex topics. And this will require a small thought experiment:
Would a tabula rasa have a purpose? No, it would not. By definition. Of course, we know that humans are not born as tabulae rasae. So are humans born with a purpose? I would argue not. We may be born with predispositions, skill in potentia, and other heritable traits, but I do not believe people are born with a purpose. I take that position in no small part on definitional grounds, as I define purpose (roughly) as the sort of positive impact one wants to make on the world. Without having been in the world to experience the problems, challenges, and opportunities facing people and communities, a human cannot have a purpose.
That is, purpose is not solipsistic. I could imagine a person living on an otherwise uninhabited island and living a purpose, but only if that person had already encountered humanity and learned something of the human struggle.
So I think purpose is not something that strictly lives inside of a person. I think it must emerge from a brush with the human experience—of others. Once that happens, I do think the purpose can "reside" in a person. But even there, purpose is not (and should not be) immutable. One can find a new way to create positive impact and alter ones' purpose. Sometimes that may be necessary as the human condition evolves.
Finally, I completely agree that many—perhaps most—people are unaware of their purpose and/or unwilling to live it because, as you say, "We're just often too busy worrying what people think of us to notice it's there."