Depending on the type of document it might or might not be the right approach.
Like ConsulNetwork commented below, sometimes a first draft goes through multiple reviews fast, and a 'shitty first draft' although might just be a first draft to you, might reflect poorly on how others view your work - as shitty. So the right expectation have to be set very clearly and sometimes with passing through so many hands, it's better to produce a somewhat decent draft.
Another type of document that I would not bother with doing shitty first draft is project staffing plans LOL The last CR I have to created went through 6 different drafts and it was just a waste of efforts (each version). This was mainly due to management wanting to see something but without providing the correct timelines.
@HubertConsultin thanks for the reply - the point of you and ConsulNetwork is really interesting. When I do a first draft of a document, I don't share it or give it into any official "review level" - I only use it to improve the first version that I share with anybody else. I'll take up your feedback and clarify that in the post.
On another note: Do you have formal rules for how documents are reviewed and by whom in your projects?