People can be independent and still have ideas congruent with a single political party.
You vote. Does that mean that you aren't independent of that political party?
It is no different with groups of people.
Yes, but what are the odds?
It'd be an awful lot more believable-when-seen if we didn't have a special built-in bias mechanism whose function is to create exactly that situation.
Besides that, both types of boards can have a variety of unofficial/official power structures, consideration of the welfare of non-constituents/non-shareholders, and varying degrees of competency.
This isn't to say that any type of governing body has arbitrary properties. Predictable inequalities exist.
Governments can be investigated by legislators, executive agencies, and courts, as well as subject to document requests (e.g., FOIA in the US), and journalists. Governments themselves don't typically operate as an advertiser in the press, hence, putting purse control over coverage.
Governments are typically more long-lived, and hence have longer-lived planning and concern horizons, than corporations.
It's not perfect, but political power is a check on corporate or financial power, one of several.
> At think tanks like Brookings, the majority of reports and events...have no obvious link to corporate donors.