I'm really happy to see any new initiative to improve games scripts. After working on 80 games projects (and in other media) may I suggest the following ways are the best ways to improve your games scripts before you get to reading them with actors...
1. Hire a writer.
Someone who can demonstrate an interest, enthusiasm and talent for writing. Not 'just' Bob your mate. Bob could be great, but someone who has taken writing courses, written scripts etc. will be way ahead of Bob, unless Bob has done some of these things too.
Bob might be great, but would you let Bob do the programming, budget the game, or create its art without proven skill? So, don't do that with writing.
Bob is cheap? Hiring expertise generally saves money as well as improving quality. Remember, Bob can't fix it if he doesn't know what he's doing.
2. Hire the right sort of writer.
Some writers write comedy, others tragedy. Some are great at text, others at cinematics. Hire for the sort of writing you need for your project, or you might not get what you need. Just like programmers, artists, etc, there are different sorts of writers, so choose one who matches the genre and style of writing you need. Or talk to them about the sort of genres and styles that will fit your game. Find writers that fit your style and your team.
3. Listen to them.
I know, I know, that sounds crazy... but if you're hiring someone for their expertise, you only get that by listening. So, give them a seat at the table, input into the project and ways to have their ideas realised. don't just hire them to pretty up the project.
The writer has to listen to you too, so give the writer a chance to converse with the team on a regular basis (as well as time away to hide in a corner and write). If you use remote writers (a great way to find writers for your team), arrange site visits.
4. Have clear lines of communication.
Who gives notes to the writer and when? It should not be a free-for-all and the writer needs a measure of choice of how to apply notes (back to expertise). Ensure one producer is part of the narrative team.
If you are putting writing into the game, give it a proper budget so that you can pay not just for the writing but the animation/toolsets/VO etc that will be involved in getting the story into the game.
Most projects try to do too much story without the right resources. Don't do that and everyone's life will be much easier.
6. Give the writing team proper time to do the work.
Writers are often hired too late, or given too many changes when design alterations are made (often because they aren't given a place to warn of issues). Good writing takes time and a measure of stability. So, does good anything
"We have booked a recording session for next week" happens too often... Guess what? That ends up with lower quality scripts in games.