This has absolutely nothing to do with design documents.
As a developer you are not supposed to start working on whatever you like, you need to follow the big plan.
Is it really like this in google with developers wasting their time just writing design documents on whatever they like that will obviously be rejected rather than doing the work that is expected from them?
I don't think this is very productive with people working on a whim instead of following a plan in a coordinate way.
You're exactly correct. Outside of 20% time, I can't see that example being really realistic (at least not at Google for an android launcher, there are other companies where I've heard of 10+ competing internal libraries for the same problem developed by different teams).
In the case that you're asked to implement a project from 'above', they're still useful, for practically the same set of reasons (maybe minus #5, but plus 'other people who you impact can provide feedback to reduce future friction')
