Well, just because modern Ashkenazi may all be descended from 350 individuals, does not necessarily mean that the population ever got as low as 350 at any time. There could have been many more alive at the time of the greatest bottleneck, but all their descendants were wiped out subsequently, in one or more different events.
Exactly! There’s no reason at all to believe that only breeding 350 individuals existed.
Population bottlenecks happen all the time and have happened in the past.
Let’s not forget that there have always been some number of European Jews who have converted to Christianity and whose offspring were subsequently “lost”. Some converted through conviction but many as a practical matter.
Just like today, not all members of a faith are equally strong in it.