In all seriousness, as a college student majoring in Chinese linguistics & a native speaker, all of the above explanations don't make sense to me. I think there’s nothing grammatically wrong with this sentence, which means it is simply a bad question.
I'm going to guess that 人民 ("the people") should be either 民众 ("the
populace") or 老百姓 ("ordinary people"). I also feel it omits
grammatical particles (的 地 得), but maybe that's okay. I don't really
人民生活水平 is a common expression in Chinese, and the usage of 人民 here is correct. This sentence, however, does not omit any grammatical particles (的) either, since it's feasible to use a sentence to serve as a subject. "國語的推行需要大家的努力"(literally The implementation of the Mandarin language requires the efforts of all) and "在全社會推行國語需要大家的努力" are both grammatically correct.
In summary, 一切 is an abstract term but this usage is too concrete.
Please pay attention to this: "一切 and 所有 may not have a scope." This means that
it's fine even if it has a scope. E.g. When preparing for a dinner, people often say "可口的蛋糕、浪漫的蠟燭，一切都已準備好了。" Nevertheless, to some extent it's also an abstract usage since the things prepared includes but not limited to the two preparations (cakes and candles) previously mentioned.
Technically, there's no grammar or spelling mistake, But when the
sentence only listed two points, using 'both' is better than 'all'.
This is exactly the point. There's no strict regulations on the usage of "both"/"all" in Chinese. Some might say one way is better than the other, but basically it's due to the impact of English grammar. It is grammatically correct to use "一切" to address two things only, though it might not meet everyone's sense of language.
In summary, I see no reason to blame that dude in the video. This is merely a bad question.