Yes, companies have complex problems to solve, that require more than a few days or weeks of work at a time. If you want to just wake up every day and decide that you need to work on a totally new project, you're not going to be very useful.
Lots of people have a wide range of interests, but you pick one of the many areas that seems interesting and that you can contribute to, and you work on that for a while and try to make a difference. "Adopting a role when necessary" is not that different from slotting in to a team, since you can often change teams every few months if you want to, or even work across teams.
What you describe is a hard role to define and hire for, but it's a role that many engineers create for themselves by being both great problem solvers and willing to do what is most valuable for the company. Consider that maybe you haven't gone in with the right mindset to make it happen for yourself.