The biblical story (actually there are several versions of the story in the OT) of the exodus is legendary but also it likely draws on semi-historical elements. The same is the case with the late Mesopotamian legends of Semiramis and the late Egyptian legends of Sesostris and Osarsiph (both of which likely influenced Jewish traditions about Moses); these three legendary figures are loosely based on historical figures. The biblical story likely draws on a range of originally independent and heterogenous traditions and combines them into a single national epic; the biblical story has connections with the Hyksos expulsion at the beginning of the 18th Dynasty, the Typhonic myth dating much earlier to the Middle Kingdom but popular later on as well (= the plagues story), the situation involving Asiatics in Egypt during the 19th Dynasty, the Egyptian Osarsiph legend which was inspired by the historical Akhenaten in the 18th Dynasty, and the upheaval and collapse of Egyptian hegemony that occurred in the 20th Dynasty (at the end of the LBA). Many traditions are thus telescoped and elaborated in a single epic story; the story of Nimrod in Genesis similarly draws on several figures widely separated in time and space. The legend of Semiramis is ultimately based on a historical figure, the Assyrian Queen Shammuramat of the 9th century BC, but attributes to her deeds actually done by the Babylonian king Nebuchadnezzar II in the 6th century BC. All of this is typical of legends (compare, for example, Arthurian legend).
There was not a single time when Asiatics ancestral to later Israel came to Egypt. There were population movements back and forth for centuries. The biblical notion of the entire nation of Israel being descended from a single family had a unifying political objective, but it isn't history. The swelling population of Semites in Egypt during the New Kingdom was not due to a single family growing to a whole nation but because of successive deportations of captives from military campaigns in Canaan to Egypt, as well as normal immigration during the time when Canaan was an Egyptian province -- leading to a significant population of Egyptian-born Asiatics. The Hyksos came to Egypt in the 18th and 17th centuries BC, and these people after being expelled from Egypt contributed to the population of Canaan. The Semites living in Lower Egypt during the 13th and 12th centuries BC (including those descended from exiles taken by Ramesses II and Merneptah) who left Egypt during the political turmoil of the 20th Dynasty also likely contributed to the population of early Israel. There are no early records of an "exodus" along grand biblical lines, although later writers associated Manetho's description of two major exoduses from Egypt (the expulsion of the Hyksos at the outset of the 18th Dynasty and the expulsion of "defiled" priests in the 19th or 20th Dynasty) with the Israelite exodus. Many records exist of Semites taken captive into Egypt, although these are not of Jews or Israelites per se -- with the exception of the Israelites exiled by Merneptah. These were taken captive in 1206 BC when the people of Israel was already present in the highlands. The literary and historical evidence persuades me that if there was a historical basis of the exodus (other than the older Hyksos traditions), it occurred in the twelfth century BC when Israel was already in place in the highlands. There is much evidence for "Israel" in the land of Canaan long before this time, e.g. the reference to Israel in the Merneptah Stele, the reference to "Asher" in Canaan in monuments of Seti I and Ramesses II and "Zakkur" (= Issachar) as the name of a Canaanite district as early as Thutmose III. The names of other tribes reflect their origin in situ in Canaan, e.g. Benjamin ("sons of the south", the southern location of Benjamin), Ephraim ("fertile soil," as a key agricultural region), Naphtali ("height", as it is in the highlands), etc.
I think it is likely that the exodus traditions conflate various legends and reminiscences of the LBA, and among these are possibly memories of Akhenaten. However the resemblence is closest with late Egyptian legends inspired by Akhenaten (the Osarsiph tale and variants thereof), not the historical pharaoh. And there is much in the narratives that demand a later date, e.g. the thirteenth and twelfth centuries BC. The references to Pi-Ramesses and the "land of Ramesses" reflect the situation in the 19th and 20th Dynasties, not earlier; Pi-Ramesses hadn't been built yet, much less was there yet the pharoah that was its namesake. P. Leiden 348 from the reign of Ramesses II refers to 'Apiru laborers in the construction of Pi-Ramesses, P Anastasi V refers to runaway Asiatic slaves escaping from Pi-Ramesses to beyond the fort of Tjeku and the Migdol of Seti I (cf. the Migdol of Exodus 14:2), and Papyrus Anastasi VI makes reference to Pithom where Shasu from Edom sought water for their flocks. The proximity of Pharaoh's palace with the foreign settlements in Lower Egypt fits well with the situation in the 19th and 20th Dynasties (as well as with the 15th in the Second Intermediate Period), and not the 18th. Also the reference to the Philistines in Exodus 13:17 points to a time in the 20th Dynasty, and this is not mere anachronism since the Philistines also figure in the very early Song of the Sea (Exodus 15:14-15), and the references to the Philistines, the "chiefs of Edom", the "leaders of Moab" and the "people of Canaan" reflects the situation in the twelfth century BC, as does the allusion to the Sea Peoples in the Balaam oracle in Numbers 24:24. The tribe of Dan was also likely one of the Sea Peoples engaged against Egypt (the Danuna/Denyen/Danaoi, cf. Judges 5:17 and the location of Danite settlement next to the Philistines in Joshua 19 and Judges 14-16, and the implication in the Blessing of Jacob that Dan was originally not among the tribes of Israel), and the priest of the tribe of Dan at the time of the relocation to Laish is named as Jonathan the grandson of Moses (Judges 18:30), which more felicitously points to a timeframe in the twelfth century BC for Moses than earlier, especially since Aaron's grandson Phinehas was also still alive just before the accession of Samuel (cf. 20:28). The toponym may`an Mênepto a ch "spring of Merneptah" in Joshua 15:9, 18:15, marking the border between Judah and Benjamin, is a probable reflection of Merneptah's campaign (why else would a place in Canaan bear Merneptah's name?), and thus the conquest narrative includes a place name that could not have been named before the reign of Merneptah (late thirteenth century BC). There is also the curious statement in Exodus 23:28: "I will send the hornet ahead of you to drive the Hivites, Canaanites, and Hittites out of your way" (cf. Deuteronomy 7:20). Some have suggested that this is a reference to the Egyptian military (cf. the use of the hornet in Egyptian hieroglyphics as a sign for the king of Lower Egypt), with the implication that the "Israelite conquest" is preceded by destructive campaigns by the Egyptians in the land. This again fits the picture with the twelfth century BC, not an earlier time.
The main thing is that the biblical story assembles traditions from different tribes into a single, linear narrative -- such that the whole nation participates in the saga of exodus and conquest, with the descent of all the tribes (or rather sons of Jacob) into Egypt occurring at the same time, the escape from captivity occuring all at the same time (therefore requiring improbable numbers of Israelites wandering in the wilderness), and conquest of all the cities of Canaan occurring in one fell swoop (as it is presented in Joshua). And this is why no one will never find evidence for one single catastrophic exodus from Egypt in history and archaeology, or evidence of a single devastating "conquest" of Canaan. In the case of the latter, there is no doubt that the various cities mentioned in Joshua were destroyed at different times in the fifteenth to the twelfth centuries BC. And the biblical data itself contains countless discontinuities and details that betray the independent origin of traditions outside the "canonical" narrative. So what we have in the OT is a conflation of reminiscences of events spanning over several hundred years across a dozen or more tribal groups compressed into a single epic story. It is natural for this to be case, as simplification is a basic process in folklore and oral tradition.
