Well Foundation in V_alpha case seems quite simple: you build externally-countable chain of subsets which simply cannot be represented as a set inside the first model of ZFC. So the external WF is not broken because the element-relation inside the models is different, and the inner WF is fine because the chain of inner models of external ZFC is not an inner set.

In the standard case your even-numbers explanation nicely shows what goes on — quoting is involved.

I need to think a bit to say what woud halt our attempts to build a chain of transitive countable models...