Can you elaborate a little bit on what you mean by this:
"Also, most startups don't open source their core business code. Some do, and I think it makes sense for them. I don't think it makes sense for us though."
My understanding is that normally as a business the considerations here would be competitive, right? But that doesn't apply in the context of non-profits. If someone else came in and used your "core business code" to make something which was then helpful to other people, that would be a good thing, right?
Likewise, your statement about the app being specific to you and not useful to others makes sense within the normal parameters of a business considering open source. But I think the idea is not that your code would be a self-sustaining open source project because it's especially useful as code, but rather that people (for example here) would be willing to help you do your work for free because they support the mission of your organization. A sort of "in-kind" donation of their time and energy and expertise, right?
I definitely think that between your admirable mission and the prestige of your network you could certainly be getting lots of free help and resources beyond just cash donations. If you could develop a workflow (via open source or some other mechanism) to take advantage of volunteer engineer labor, you could presumably pursue whatever software goals you have radically faster, I would think.