How writing builds professional capital
Professional capital is the reputational leverage you have in your field of expertise. It's the kind of reputation that you can take with you to get better opportunities, whether that's a job, an investment, or access to interesting people.
Imagine two characters: Mack and Joe. Mack is working at a large, prestigious company but is an average PM. When he goes to look for another job, hiring managers immediately give him a chance because he comes from a big name. Joe works for a small, relatively unknown company, but is an excellent PM. However, only the people at Joe's company know this about him. When Joe goes to apply for jobs, relatively few hiring managers pick him out of the initial pile.
In what world could Joe beat out Mack? In a world that relies on the internet to get information.
If Joe starts writing about the latest products and how they are evolving, sprinkling in interesting analysis and opinions, he can start building a readership. If those readers end up being hiring managers of companies he's interested in, suddenly Joe becomes hot talent. Not only is he someone that the right people know about, but he's also constantly showcasing his ideas and talent over time, rather than in a 15 second resume skim.
Joe can beat Mack if he writes to build his professional capital.
First, writing is an essential part of the product job. You need to write product requirements, copy, strategy documents, etc. A great portion of a product manager's time is communicating to various stakeholders a consistent, cohesive vision. That requires a lot of writing because written content is how information is primarily transmitted in organizations.
Second, writing is an extremely leveragable and easy to create medium. Yes, you can create multiple types of media on the internet. But writing is effective because it is the oldest and the easiest form of creating professional content. While you could produce professional videos, audio, or images - all of these are significantly more production value than writing. Other mediums also tend to depreciate very quickly. People have a tendency to consume the latest videos, audio, imagery, etc. Writing tends to depreciate significantly less. You've probably read many old books, blog posts, and articles, but you probably don't listen to many old podcasts or interviews. Despite being completely outdated, people still read Warren Buffet's shareholder letters.
So if you write, you express both the ability to communicate effectively and you create a highly leveragable content for yourself.
Something You Own
Writing content also gives you something incredibly important for your professional journey - it gives you something you own. When you change jobs, you switch titles, names, logos. What reputation you gained in one company will not transfer to another one. To some extent, you must start from scratch, because you can't take all your reputation with you.
But when you write and publish on your own, you can. What you write about becomes your own asset, not something that is borrowed from a company. It's a good long-term asset to build for yourself.