From my undestanding a standard method of deducing relative consistency results is the following:
By a combination of the Levy reflection principle, Skolem-Hulls and Mostowski Collapse we show:
If $ZFC$ is consistent, then "$ZFC$, $M$ is transitive and countable, $ZFC^M$" is consistent, where $M$ is a new variable.
So we can assume there is a countable transitive Model $M$ of $ZFC$ form the forcing extension $M[G]$ and show $M[G]\models ZFC+\phi$
But why do we have to invoke the Reflection principle, what is for example wrong with the following reasoning:
Assume $ZFC$ is consistent, then by the Löwenheim-Skolem theorem there is a countable model $N$ of $ZFC$, taking its Mostowski transitivisation we get a transitive, countable model $N'$ of $ZFC$ and thus can construct $N'[G]$. So the reflection principle becomes superfluos? Or is there something fundamentaly wrong with my understanding of the former method in the first place?