> If they're not soliciting donations from you I'm not sure why you'd care about their federal tax status.

Well, if they portray themselves as a "nonprofit" then most people who read that will think they are a 501c3, which is almost always the case. I don't know why they don't qualify for that status (if they don't), but it's possible that it's a reason I would care about when deciding whom to side with on issues like this one.

The battle of for-profit versus non-profit comes across differently than for-profit versus Michigan Domestic Non-Profit Corporation (which for some reason does not qualify for IRS nonprofit designation).

It's not "almost always the case". It may be the case for nonprofits that people donate to, but in general there are quite a few 501c4 around, for example, and there are many others: https://en.wikipedia.org/wiki/501(c)_organization#Types