"For what it's worth, I have a bit of experience..."

For what it's worth, I have a bit of experience with Kickstarter campaigns, where a similar pattern is often alleged. The sense there is that the "last minute push", while it does happen sometimes, is not guaranteed, and you shouldn't rely on it - instead, the conventional wisdom is that you shouldn't even start your campaign until you've built a bunch of hype with the relevant people and have a strong initial push all but locked in.

This allows you to begin the campaign with a strong signal of interest and buy-in; some Kickstarters are planned/calculated to fully fund on day one and immediately start going for stretch goals. I'm not sure if a similar pattern would be effective for nonprofit fundraising, but it seems perhaps worth trying.