I honestly don't think you'll be able to get very far unless the team is already onboard with the plan to switch towards a more asynchronous culture. If they lack the motivation, they won't bother improving their writing.
The tricky part is that they might be interested in the results you promise, but still lack motivation. It's common for someone who's interested in losing weight not to be very motivated to do it themselves. They don't want to lose weight themselves, they want to "be slimmed down by someone else". You may face a similar difficulty. That's part of the reason why changing a culture is so hard.
I know that's not a direct answer to your question, but I needed this context to say this: I think whatever tricks and tips you can come up with yourself are more likely to succeed. That's because you're already familiar with the specific needs and the specific difficulties people might face when handling the most frequent and repetitive issues.
Rather than thinking how to get fuzzy improvements on people's overall writing skills, perhaps you could try to focus on suggesting specific solutions for specific problems ("hey everyone, I've noticed that when handling X people usually forget to tell p, q and z. So let us agree on using this text structure '1) p; 2) q; and 3) z; whenever handling X"). I think that, by accumulating lots of small tricks like these over time, you'll be able to go further. Going bottom-up seems easier than trying to change things top-down.
I'm not sure this makes sense to you, for I'm not familiar with your concrete situation, but I hope it helps somehow.
