We don't have a design system. What we have is a set of products built fast, for users who needed the imagery yesterday, that share a brand without sharing much else. Buttons that quietly disagree with each other. Spacing as a matter of personal taste. Several defensible ways to build a table, none of them the same. Nobody was sloppy - this is what shipping under real pressure leaves behind, and the interest on it now comes out of every sprint.
As a Staff / Lead Product Designer, you'll build the thing from nothing. That starts small and slightly unglamorous: a Figma library people actually reach for and a design playbook that settles the arguments worth settling. Then tokens with a structure that holds. Then components in code, documented well enough that nobody has to ask you how they work. By the time it's a design system in the full sense, the hardest part - people using it by default - is already solved, because you'll have built it in the open with the teams who had to adopt it.
This is a hands-on, opinionated, sleeves-rolled-up role. You'll be the highest craft bar in the domain and the person who raises everyone else's. You'll spend as much time with frontend engineers as with designers, quite possibly in their repo.
Real ownership. Very little existing process to inherit. If that sounds like a warning, it isn't. It's the job.