Sean, NO, you don't need a business plan. And the reason why is very important, but it might sound strange to you: there is a difference between a company and a startup. A startup is not a smaller version of a large company. A startup, as Steve Blank describes it, is a "temporary organization in search of a business model". This is a different scenario than if you were just buying and opening, say, a McDonald's franchise. With the franchise, you don't have to SEARCH for how to run it. That has already been worked out. The customer base is there, the products, the channels, etc. All you have to do is BUILD and EXECUTE.
But a startup doesn't really know what it is or finally will become until it hits solid product/market fit and it has identified and proven exactly what problem it is solving, that its solution is correct, who the exact customers are and that they will buy it (among another things). And you will likely pivot in small or large ways that will CHANGE your plan as you go along.
This is why a business plan doesn't make sense for a startup. Business plans are made to simply be executed. A startup needs a looser, more fluid framework for their planning. And most investors these days understand the difference and don't require business plans at all.
The normal docs you would be expected to have minimally when approaching angels would be a pitch deck and an executive summary..
For more info on the different approach to planning and executing that a startup needs to do, I really recommend an online course that Steve Blank did (it's a FREE video course) called "How to Start a Startup" and you can find it a Udacity.com
. I took the course myself and it helped a lot.