Gauge fixing question

Apologies for the cross posting but this seems to be the place to put it...

I have a question on gauge fixing and path integral evaluation. I hope
it can be answered here. Starting with say, the Maxwell action, we can
write the generating functional as

Z ~ int DA delta[ G[A] ] det M exp{ -i S }

where G[A] = 0 is our gauge condition, and det M is the Faddeev-Popov
determinant. The delta functional can be massaged into the exponential,
giving a term -1/2alpha G[A]2 in the action, which is the "gauge fixing
term".

My question is that initially we enforce G[A] = 0 using the delta
function, but once this gets turned into a "gauge fixing term", it seems
that A no longer needs to satisfy this, and classically the EOMs are not
the maxwell equations. I realize that any physical quantities don't
depend on this, but I don't see where the magic happened. Why doesn't
the gauge fixing term "fix" the gauge?

Thanks to anyone who can clarify.

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