> Given what we know about how the Egyptians recorded history, we would definitely not expect to find them writing about stuff that would have embarrassed them.

That's exactly the sort of stuff they wrote about all the time. We know about the various wars and political conflicts throughout the second intermediate period precisely because that's what the Egyptians liked documenting.

And, in particular, during the supposed time of the exodus the Egyptian kingdom was fairly divided. Even if one kingdom was too proud to write about a defeat, the others would be sure to document it.

> Look up Papyrus Brooklyn 35.1446 - it shows that Egypt held slaves with Semitic names in roughly the correct time period.

Read up about the Canaanites. They were on the uprise during this period and they are also believed to be the actual origin of the Hebrews.

> Israel being one of them!

No even according to the bible. Israel didn't exist before the exodus. Definitely not for decades and even centuries afterwards. The oldest records of the exodus are nowhere near the event. The closest record we have is around 900BCE.