Yes, it is true that you can apply now and provide the marriage proof later. Just be advised that in the interim they will be asking for proof of cohabitation for all those periods (and I agree that just flight itineraries would be a bit thin, but if you have other evidence, sure).
The issue is that they'll be doing the analysis of the common law evidence (somewhat subjective of course so more resource intensive). Do not bother with detailed explanations of plans to get married - they'll give it approximately zero weight until you have the evidence of the legally-performed marriage, and while they may look at the docs given you by the officiant, only the marriage certificate will serve as proof of marriage.
Whether it will in practice save much time depends on specifics and I don't want to guess. Given your timeline, I'd roughly say the chances of rejection before you complete marriage are pretty low, and that should at least get you the AOR so your spouse can apply for the work permit.