Great question. I think you’d treat yourself like the prototypical large company that grew by acquisition. In these situations, companies acquire vastly different products and teams and have to use the design system to align.
You face a cultural challenge as well. In my experience, one of the best ways to get orgs and divisions to respect design is to make immediate impact. Design systems are typically a tough way to show this because the efficiencies and benefits aren’t easy to see outside the teams. As you build your team you might first consider giving them tasks of challenging and innovative projects and you focus on some of the system and process work.
And as you mentioned, the rebranding itself might be the perfect vehicle to deliver on the system and innovation front.