Now finally getting back on topic, to the question of whether Baby Jesus did go to Egypt.
Once again, the theory you present is reasonable, but it is "reverse-engineered," in the sense that you (and others) read these accounts and filled in the gaps.
Now your theory is possible, or maybe the writers of these two books were reverse-engineering the story of Jesus's birth long after those events are supposed to have happened, and since they lived in the pre-Google era, couldn't check their facts together.
So your theory depends on the timing of the arrival of the wise men. Of course, we have no evidence that such men ever existed. And if they did, we have no way of verifying the timing of their visit (the Gospels don't say when they came, except at some point after the birth of Jesus).
Your theory is therefore reasonable, but neither verifiable nor falsifiable.
My claim, that the gospels can't get their stories right together, is based on just reading the texts without looking for excuses or gaps in the time-line: each gospel mentions an itinerary for travel after the birth of Jesus, they are contradictory.
Either the gospels contradict each other, or as you speculate, there is a gap in the time-line.
Which is it? I'd go with the contradiction, because Matthew doesn't mention anything about a gap, but your theory is neither verifiable nor falsifiable, so you can believe it all you like