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legal immunity from demands of the executive branch" would take a constitutional amendment; anything less can be rescinded with a simple majority vote or a judicial decision.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...