According to Rabbi Soloveitchik, the reason God responded so badly is because eating meat is actually something God abhors very much, and therefore tries to put as many restrictions on it as possible. According to Soloveitchik, one of the reasons that God flooded the world was because mankind started eating meat, which is why God allows eating meat after the flood, so that way He wouldn't have to destroy the world again. So in actuality, what happened in the desert makes a lot of sense when compared God's response to the generation of Noach.
R. Soloveitchik’s severe stance is based on the story of Kivrot ha-Ta’avah (the graves of those who craved [meat]), the tragic account of Benei Yisrael’s lust for animal flesh. In the story of Kivrot ha-Ta’avah, Benei Yisrael protest to God and Moshe, demanding meat instead of the manna that God had been supplying. Moshe prays to God and, although God is angry with the people, He gives them the meat. Once satiated, the people die as a result of a plague that God sends. In his explanation of this story, R. Soloveitchik says that God admonished Israel for their dissatisfaction with their vegetarian diet of manna and their need to have meat. Deuteronomy 12:20, in discussing God’s commandments for when Benei Yisrael will live in the land of Israel, supports this point: “And you shall say: ‘I will eat flesh’, because your soul desires to eat flesh; you may eat flesh, after all the desire of your soul.” The Torah uses the word “desire” to characterize man’s hunger for meat; it is the dominating physical desire. Hence, according to R. Soloveitchik, vegetarianism should be practiced, yet man, too desirous for meat, refuses to stop eating animal flesh.
In explaining his point, R. Soloveitchik cites Sanhedrin 59b, which
says that Adam did not eat meat, and it was only when Noah entered the
biblical narrative that meat was permitted. Commenting on this, R.
Soloveitchik states that the natural reality of Adam’s distaste for
meat became the ethical norm with the phrase, “and it was so.” R.
Soloveitchik explains, “Thus the verse concludes ‘and it was so’: the
ethical norm became a behavior pattern, an expression of the ontic
order.” The ethical imperative against eating meat becomes the
physical and biological reality of man’s world—no one would eat meat.
Yet, as the history of man continues through dor ha-mabbul (the
generation of the flood), man begins to overreach himself, to take
what is not his, including the life of another living being. Thus, God
eventually gives in and allows Noah to eat meat: “Every moving thing
that lives shall be food for you; even as the green herb have I given
you all things.” R. Soloveitchik explains, “At once the Torah began to
regulate the ‘murder’ of other lives, to restrict its practice by
complicating the procedure… ‘[the Torah succumbed to the Evil
Inclination by allowing for certain things, hence] the Torah provided
for human passions: [reasoning that] it is better for Israel to eat
the flesh of animals that are ritually slaughtered than the flesh of
animals which have perished [i.e. nevelot (the dead unslaughtered
carcass of an animal)]’ (Kiddushin 21b-22a).” R. Soloveitchik explains
that the Torah allows man to fulfill his desire for meat, but out of a
care for animal life, it complicated the process of acquiring meat.
Based on this view, your example of a spoiled child wanting more than he deserves doesn't quite fit. It's much more closer to having your child suddenly demand that you murder your family dog to feed him even though you've already given him food to live off of. It's a much more disturbing reality. Since we've been consuming meat since the flood, it's hard for us to picture it being such a big deal, but as Rav Soloveitchik points out, God has been remarkably consistent regarding His dislike of people eating animals.