It happens, as does grammaticalization, but it takes a looooooong time. And the existence of a large extant written corpus and an educated population (as it is the case today, but wasn't quite the case back then) makes it all the harder. As long as you don't have written language, you can have very disparate regional dialects and language quickly evolving. But the need to communicate across time and space seems to stabilize languages in some respects (just witness the case of English spelling, for example).

(Also, Old Norse was quite a bit closer to Old English than presumably modern English is to some of those languages in question - Finnish, especially.)