<i>sponsoring an open-source project so that it is well maintained?</i><p>Invalid assumption.<p>There is no guarantee of any kind with open source --- not for maintenance or direction or control or support or schedule or anything. Open source makes demands but offers no real promises in return.<p>If this is really a "critical" app and you need better control, I'd say you have no sensible/realistic option but to either built it yourself or at least maintain your own version of it. Lots of people have figured this out the hard way.
Is the proposed software something that creates a competitive advantage or is foundational/plumbing?<p>I'm not going to fund an open source project that would represent an even fleeting competitive advantage. On the other hand, I'm not going to pay to build an in-house CMS. If I had the money would consider spending it to support some open initiative. The rub I see though is that most startups don't have the runway to throw discretionary spending at open source.