I've worked with a number of writers over the years and many have had their "preferred" method of edits. Most are willing, or at least understand, that compatibility is an issue and a common workflow is necessary so everyone isn't unfairly burdened with edits/changes.
Overall what works depends upon how extensive the edits may be.
If the design files are Illustrator files I assume they are single page or single signature files. I would never use Illustrator for a multi-signature project. If you are, your workflow may need adjustment. Or perhaps it means you have to deal with PDF commenting due to the overall nature of the pieces (mobile screens, or some other common multi-file set up).
As @user8356 points out in comments... for single-signature projects using Acrobat and the commenting tools within Acrobat to denote changes needed is often the easiest, most trouble-free, method assuming changes are relatively minor. Using Acrobat's commenting has several benefits. Any notes can be directly copy/pasted from Acrobat. All comments are listed, with the ability to check/uncheck them, so nothing gets overlooked.
For extensive changes across a multi-signature document it's often easier to provide a text or Word file for the writer to edit. That does mean the designer needs to export, copy/paste, or save the copy to a text/Word format. You can use Acrobat to save as a Word file in many instances. The writer then edits the text, sends it back, and the designer copy/pastes text back to the design file. A PDF is then sent back to the writer where they use Acrobat commenting to denote any inconsistencies.
This may require that the writers become familiar with Acrobat commenting, but it is a worthwhile learning venture in the long run.
I also have one client that prints PDFs, and uses a pen to mark up changes, then uses a copier/fax/email machine to email be back a (scanned) PDF with the changes. This is by far the worst workflow, but it does work.
In the long run there's no magic shortcut I've found. Writers often don't have the (expensive) software that designers have. They just don't have need for it so why spend the money. And the design software doesn't anticipate edits without itself. So, inevitably it means one has to work around the compatibility issues.