Kitchen demonstrates the problems with from the archaeological data itself. The main problem according to him is demographic. The population in Palestine was very small due to Hittite and Egypt oppression. Also, the amount of population increase is very, very large. We do not have a doubling of the population, but between the 13th and 12th century BC there is a fivefold population increase. Now, that's not because Israel started making lots of babies, but because they gradually settled down. Nomad don't leave any trace of themselves.
A fivefold increase can't mean merely that Canaanites who are non-nomadic are move to another area where they're non-nomadic again. We'd get a shift in population if that was the case, not an increase. In other words, they'd leave material culture traces of themselves in both places. Or perhaps it would mean that their infant mortality rate went way, way down, while they magically started having a lot more children. In other words, not very likely.
Kitchen also points out the absurdity of the idea that they were trying to escape the city-states on the plain to gain freedom from heavy taxation. If so, then they would simply have been followed into the hills by the tax-gathers. Jerusalem and other Canaanite city-states certainly controlled the hill country and could tax people there just as well as they did on the plain.
From my own person study, I would also mention some other problems with the thesis.
1. If the exodus was only a historical memory for a very small part of the population, why was there no continuing folk-memory of escaping the city-states and only one of escaping Egypt? After all, why would one be inclined to believe one had been a slave (something fairly shameful in the ANE), when one hadn't been?
2. Where did the Hebrew language magically come from? Why wouldn't the Israelites (if they were displaced Canaanites) just speak Urgaritic like everyone else in Canaan? Instead, we have a very, very different language that has arisen, not just a new dialect.