It's one thing to describe the Anyang culture or to construct a literary analysis, but does anyone actual belief the myths? No one goes around trying to prove that the Amazons were real, right?
This is a loaded question that misleadingly implies that there's a scholarly debate over the existence of the Zhou dynasty. There's isn't. The Zhou dynasty is well within the recorded history of China, and no serious historian questions its existence. Many of the classical texts of Chinese literature and philosophy were written during the Eastern Zhou dynasty (the political authority of the Zhou royal family had declined by the Eastern Zhou, but the Zhou kings were still in power as nominal figureheads). It's as real as the Peloponnesian War.
The current consensus among historians is that there was a Shang dynasty. The Shang dynasty used to be questioned by the Doubting Antiquity School of Chinese historiography in the first half of the 20th century, but the discovery of oracle bones has convinced modern historians that the dynasty did exist. The oracle bones were used by diviners for conducting sacrifices honoring past kings, so all of the kings of the Shang dynasty are named. The reconstructed Shang dynasty kings list is basically identical with the Shang dynasty kings list in the Han dynasty historian Sima Qian's Records of the Grand Historian.
The existence of the Xia dynasty is still up for debate. Some archaeologists tie the Erlitou culture with the Xia dynasty, but as of yet there is no definitive evidence linking the Erlitou culture with the Xia dynasty. The "myths" that the OP is referring to is likely the legends associated with the first Xia dynasty kings, such as how Yu the Great saved his people from a great flood.
The only period of traditional Chinese historiography that is regarded as completely mythical is the era of the sage kings, or the "Three Sovereigns" and "Five Emperors". Sima Qian included the sage kings in his Records of the Grand Historian, but no modern historian believes the story that Fu Xi was progenitor of all human civilization.
TL;DR No one doubts the existence of the Zhou dynasty. The existence of the Shang dynasty used to be questioned, but because of archaeological discoveries, the modern consensus is that the dynasty did exist. There is still an ongoing debate as to whether the Xia dynasty existed. The only completely mythical period of traditional Chinese historiography is the reign of the Three Sovereigns and Five Emperors.