Before trying to answer this, I want to push back on the idea that most other regions don't have any written source, or at least one that historians can tie to archaeological evidence. My specialty is Middle Eastern history, so I cannot speak directly on every case here, but people have used the Homeric epics, the Vedas, the Avesta, Beowulf, Arthurian legends, the Icelandic Sagas, the Shahnamah, the Qur'an, and Mesopotamian myths (including the Epic of Gilgamesh) to complement archaeological evidence. I'm not saying that some (or any) of these texts have been as important to archaeologists as the Bible has been, but to say that most regions don't have ancient written sources is a little disingenuous.

That said, your question raises an important issue in Biblical archaeology, and one that has influenced the field since its inception. How much of the Bible (and I mostly mean the Hebrew Bible/OT here) contains reliable historical record, and should that record be considered "evidence" when there are no corroborating sources to be found? There is no one easy answer to this question, and it depends on the particular context of different archaeological sites as well as the section of the Bible that one is referencing. In some cases, the Bible has been an important corroborating source for other evidence (or the evidence has corroborated the Bible, however you want to interpret that), but in other cases, the Biblical narrative has influenced historians to interpret certain finds in accordance with the Bible, even without clear evidence that the two should be connected.

In order to understand why it is difficult to use the Bible as a historical source, we have to understand how the text itself was produced and preserved. One of the first modern - and probably the most influential - attempts at biblical source criticism was developed by Julius Wellhausen in the late 19th century. He created the "Documentary Hypothesis," theorizing that the Biblical Pentateuch (AKA the Torah, AKA the first 5 books of the Bible: Genesis, Exodus, Numbers, Leviticus, Deuteronomy) was compiled from multiple documentary sources by multiple "redactors" over the course of many years. I speak only in the broadest terms here, as the specifics of Wellhausen's hypothesis have largely fallen out of favor. However, the idea that the Bible is a compiled source, created and preserved by multiple editors who each had their own historical contexts, has persisted, and now underpins most of biblical source criticism. The Documentary Hypothesis as related to the Pentateuch itself has undergone something of a renaissance in recent years (particularly at the University of Chicago, under Jeffrey Stackert and Simeon Chavel), but for now it is just important to understand that the Bible cannot easily be used as a single source.

With that in mind, we find a number of interesting contrasts between the books of the Bible in terms of their historical "reliability." The first five books seem to be almost entirely mythical, with the exception of some details about the Hebrew priestly tradition that can be derived from the "edits" (c. 6th or 5th-century BCE) to the original narratives (c. 10th-9th c. BCE). There is no evidence of any historical people named Abraham, Joseph, or Moses, for example, in the archaeological record.

However, in the 1940s, another German scholar named Martin Noth hypothesized the existence of a source that he called the "Deuteronomistic History." The name is a bit misleading, but Noth explained that the DH was a single text, written in the 6th century BCE, that spanned the books of Joshua, Judges, Samuel, and Kings (but all based on the theology of Deuteronomy, hence the name). The author's purpose in writing this text was - according to Noth - to help the Israelite community cope with the fall of Jerusalem and the Babylonian Exile. The DH theory was then revised by Frank Cross, who established that the text itself was first written in the 7th century BCE, and then updated by Noth's first proposed author in the 6th century.

The Deuteronomistic History offers the first evidence that can be corroborated by archaeology, and this is where we can return to your original question. The book of Joshua, for example, describes the collapse of the walls of Jericho during the conquest of Canaan by the Israelites (Josh. 6:1-27). In the 1930s, a British archaeologist named John Garstang excavated Jericho, and he found remnants of fortifications that he hypothesized must be the ones from the Biblical account. Consequently, he dated them to the late Bronze Age (~1400 BCE). However, he was wrong. In fact, he was so wrong that when Kathleen Kenyon went back to Jericho in the 1950s, she found no evidence of Bronze Age fortifications, and ultimately showed that Garstang's walls dated to the Neolithic period (~8000 BCE). In this case, Garstang found what he wanted to find - support for the Biblical narrative - even when his evidence did not support it. Kenyon also found that Jericho was indeed destroyed, but well before the Late Bronze Age, and at the time of Joshua the site was unoccupied. This suggests that the biblical narrative was not describing the original arrival of the Israelites in Canaan (in theory, ~1400 BCE), but rather was describing Jericho as it existed at the time the DH author (7th or 6th century).

So having the Bible as a source has, in some cases, influenced scholars to misinterpret data and make conclusions that are not supported by other evidence. But it has also proved invaluable for providing context to other archaeological finds. These finds include (but are not limited to): the Merneptah Stele, the Tel Dan Stele, the Black Obelisk, the Mesha Stele, Sennacherib's prism, and reliefs in the Assyrian palace at Nineveh.

Flinders Petrie discovered the Merneptah Stele in 1896 while working in Thebes. It dates to 1208 BCE, and describes a number of places and peoples who were within the Egyptian sphere of influence at that time. One of these peoples is called "Israel," which makes the Merneptah Stele the earliest archaeological evidence that refers to the Israelites. Notably, this stele was created several hundred years after the supposed events of Exodus, and does not support the idea that the Israelites left Egypt and went to Canaan (it also doesn't explicitly disprove it, though). The stele is now in the Egyptian Museum in Cairo, you should drop by if you're in the neighborhood.

The Tel Dan Stele, discovered in Israel in the 1990s, is an Aramaic inscription describing people under the control of Aram-Damascus. It's been dated to the second half of the 9th or first half of the 8th century BCE, and contains a reference to "the House of David," a political dynasty of some kind. This is the earliest extra-biblical reference to David. Without the Bible, which follows the story of David throughout the book of Samuel, we would have no knowledge of who this stele was referring to, and the House of David would be just one of many groups who we know the Aramaeans conquered in the Iron Age.

(There is some controversy over the dating of the City of David in Jerusalem and the lives of David and Solomon, but I do not have the books I need to properly source that. Suffice to say, the Bible has complicated the process of dating a number of finds, including a large palace-like structure near Jerusalem and a bunch of pottery from Megiddo.)

The Black Obelisk, as it is now known, is a stele constructed by the (neo-)Assyrian king Shalmaneser III in the middle of the 9th century BCE. It depicts a number of foreign kings bowing or bringing tribute to Shalmaneser, and conveniently provides Akkadian cuneiform captions that tell us who they are. One of these kings is "Jehu, son of Omri," who we know from the Bible (2 Kings 9-10) as one of the kings of the northern kingdom of Israel. This depiction is the oldest surviving picture of any Biblical figure. However, Jehu was not the son of Omri, and according to 2 Kings he actually overthrew Omri's dynasty. If we only had the stele as evidence, we would not know this, and we are thus forced to question whether Shalmaneser knew something about Jehu that was edited out of the Bible, or if he simply did not have detailed knowledge of Israelean politics. The obelisk is now in the British Museum, but there are replicas in other museums, including Chicago's Oriental Institute.

The Mesha Stele (AKA the Moabite Stone) was discovered by Frederick Klein in Jordan in the 1860s, but it was later destroyed and most of the pieces were lost. Luckily, a French archaeologist named Charles Simon Clermont-Ganneau managed to acquire a squeeze of the stele before its destruction, and it has been reconstructed based on that squeeze and the remaining stone fragments. It describes a conflict between the king of Moab and the "House of Omri" (i.e. Israel) in the 9th century BCE, and the story parallels the Biblical account of what seems to be the same event in 2 Kings 3; but the two accounts differ a bit on who "won" the conflict. It also contains the earliest extra-biblical reference to the name of the god of the Israelites, written in Moabite as YHWH. So here we have two versions of a historical situation - one Biblical and one not - and comparing them reveals some biases in both.