Most definitely. When a culture is lost so is it's language and way of life. We are still learning how different languages originated. To let a language die would be a loss of information on how people migrated from place to place. Also, indigenous cultures have a special knowledge of their surroundings; plant and animals. There may be a plant they know of that could cure cancer. The people may have an immunity to a common disease. But most of all, because it's not like us, it should be cherished, not be allowed to die out. A culture has a historical nature and like an old monument needs to be preserved.
With the corruption of government being a major issue today there's no way of knowing whether the government will or will not actually help the people. Wi th their society being at a vulnerable state the government is able to easily manipulate and exploit the people which means they're really harming them, rather than actually helping
While the creation of (an) culture is inevitable when humans exist, it is a thing controlled and governed by the (majority of) people. Culture is destroyed, created and (constantly) altered by the people. Should people choose to adopt a different culture, that culture will simply disappear. What's the problem? To actively "protect" a culture is ridiculous. Now when people are forced to assimilate, that's a problem. The same goes for monuments, areas of land, and other physical objects that represent cultures - actively destroying it (bulldozing, etc) or protecting it (from environmental factors*) is stupid.
*If some random, non-governmental organization wants to protect and/or restore them, that's fine.
In essence, my problem with this theoretical dilemma, isn't protecting of cultures itself, but rather that a government is supposed to do intervene.