I'd say there's not a single correct answer for everyone on this question, but I'll put out my two cents.
I see three general tracks through work life (VERY broad generalizations to follow). First, and pretty common, you do stuff, develop knowledge, grow seniority, start moving into management. Second, you're a worker bee. You're happy as a worker bee. You do stuff, develop some knowledge, screw seniority, I'm staying where I'm at. Third, you're just not that into the concept of managing people, but, you do stuff, develop knowledge, grow seniority, but now I want to be a leader (without that messy people stuff). Technical leadership.
To be a technical lead, which is a less common path, you absolutely must develop communication skills. Those skills need to include public speaking (so you can sell the higher ups on a good idea, teach your team & peers, etc.) and writing (proposals, documentation, even articles for publication). Not every organization can support a technical lead like this, but many can. It's the path I found myself following as I did develop seniority, but had ZERO interest in becoming a manger (I'd actually moved into that role early in life and found I hated it), so I worked with my boss, their boss, and members of the team to develop a technical leadership role. The need to be able to communicate, clearly, efficiently, and effectively, was probably more important than my technical knowledge.
So, if you want to be a worker bee, you can stay there and you really don't need to sweat much these kinds of skills. If you want move up, management or just leadership, you need to enhance your ability to communicate, so yeah, develop writing skills.