Usually a tech writer is also some sort of PM role that gets to chase developers and get them to explain what's in their head. Sometimes you have to have such a person on board.
I once did an internship that involved chasing down senior devs and generating documentation for them (needed for FIPS certification). It was a great way to learn about the tech stack. Suffice it to say I didn't have any PM role at all.
In my opinion, you shouldn't ask technical writers to manage projects anymore than you would ask engineers to. Both jobs are complex enough without adding an entirely different full-time job on top of the work.