Communication in general is helpful. Many incredibly skilled technicians are severely held back by the fact that nobody understands what they are doing and how impressive it is.
On the other end of the spectrum, most truly famous people are not just good at what they do, but also good at creating a sort of myth about themselves, or else has had a friend who loved talking them up. Learning to take what you've done or learned and spin a compelling story around it is an amazing life skill that absolutely will get you ahead.
You can absolutely take this too far and become a narcissistic charlatan -- all talk and self esteem -- but this is extremely far away from the personality of many engineers so I don't think that's an immediate concern.
I'd say being a good communicator, an "effective communicator" has the most impact. Not only writing well, but being able to enter and participate in C-Suite strategy conversations as a peer. Not dominating, not "telling them the truth", but being a peer-wise political player in the political theater that actually runs the company. That is what comes after staff engineer, it's VP of technology, CTO and so forth. That game is all communications, but as a former developer your role is getting them to understand the realities of the tech teams' efforts and successes, and the required maintenance to see it continue.