The big distinction between the two of them is that donations to a 501(c)(3) charity are tax deductible and therefore may not be used to unduly advantage any commercial entity, while donations to a 501(c)(6) are not tax deductible and thus are unrestricted.
If it's a 501(c)(6), then the foundation will serve its biggest donors first and foremost. In theory, it may also serve the public good — but only to the extent that it is in the interest of those donors.
It is true that it is more difficult to get funds for a 501(c)(3) than a 501(c)(6). That forces the organization to operate lean and constrains possible initiatives (e.g. funding development), but also serves as a buffer against influence.
Paradoxically, a maximally independent organization may serve the interests of the long tail of potential corporate donors, because they don't have to worry that the biggest player in their space will capture it via pay-to-play.
(For those not up on the acronym, BOD stands for Board Of Directors in this context.)
The most important thing is ensuring that the community has representation on the board and other governance structures. There are some organizations like the Apache Software Foundation that do an OK job at this, there are others that don't even offer projects/maintainers seats on the board.
There are also organizations like the Linux Foundation and Eclipse Foundation which essentially act as "Foundation as a Service" and host multiple foundations in one with different governance structures.
https://www.linuxfoundation.org/blog/2020/05/building-a-succ...
https://en.wikipedia.org/wiki/Donald_J._Trump_Foundation#Div...