I would prefer a for-profit one, but due to many of the concerns you point out about compliance (and that you can no longer get donated labor), I don't think you can make it work (you end up spending too much on compliance and management relative to the size of the fund). Amortizing the costs across a national fund substantially changes the incentives and control structure -- which replicates many of the problems I have with current investment mechanisms. (And you'd still run in to issues where every state has special snowflake rules about certain things.)
The reason is I think you really need to focus on localization of such funds to make them effective -- metro scale ones. You can probably make the financials work by simply eating the overhead on that scale, but not on one trying to turn a profit, but paying for labor. Losing 17% of your investment every year to overhead is simply too rough, because you'd need to generate 20% returns.
You could do it, but it would substantially change the nature of the investments you could make and the size the fund would have to be to operate. There's a certain charm to funds in the 1-10 million dollar range making small investments in 10-100 businesses. (And I believe helps combat some of the problems involved with corporations, by intentionally distorting the market by "losing" money purposefully and recouping the value from the social change.)
I was (obviously) unaware of CDVCA, but even there, it seems like they're mostly a broader investment scale and set up as a traditional venture fund (ie, looking to connect traditional investors with high growth potential ventures) as opposed to operating a continual flux of small, local businesses at (likely) negative returns funded by the local populace at large. (I poked around the NYC one, for instance.)
That said, they likely have worked out some of the legal hurdles involved.