>You can establish institutions that are elected directly by the people, and/or are granted strong legal immunity from demands of the executive branch etc.Any example you could name would be highly contested as to validity of the claim.
Specifically, "elected by the people" doesn't grant any immunity by and of itself; you would need a huge public protest or other such expression of public will. You may still be ignored by the media, which prevents you from getting anything done[0].
On the other hand, "strong legal immunity from demands of the executive branch" would take a constitutional amendment; anything less can be rescinded with a simple majority vote or a judicial decision.
>But this doesn't mean that private, feudal and corporate power is the only other option.
You would need to create a very strong legal shield around such institution. Right now the government (particularly the judiciary branch) considers every private contract to be in its purview, with ability & willingness to void, add, or rewrite any and all clauses it considers wrong. Even institutions like marriage are subject to retroactive change of the original terms&conditions. Even the media, with its oversized influence through gatekeeping the political parties & candidates, is still bound by various regulations regarding falsehoods, besmirchment, etc., while at the same time tied intimately to their insider sources. About the only exceptions I can think of is the internal dealings of the churches, and the diplomatic matters [1].
On the other hand, any NGO is subject to the wishes of whoever finances it. Whether it is public donations or corporate sponsorship (or outright corporate structure); whoever holds the spigot of the money gets a say. Another important say is held by the media; with ability to either prop up or sink any NGO or public figure.
As of recent, the social media mobs seem to be gaining the gatekeeping ability just as well.
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[0] quick examples of media keeping mostly quiet: weekly Yellow Vests protests, Paris, France, 45+ weeks; annual March for Life, Washington DC, US, 45 years.
[1] bit of a stretch, but also the https://en.wikipedia.org/wiki/Sovereign_Military_Order_of_Ma...