The point is that during the ordinary phase of the Big Bang expansion, the difference $|\Omega-1|$ was rather dramatically increasing with time. Today, we know that $|\Omega-1|\lt 0.01$ or so. If we use the cosmological equations to reconstruct what $|\Omega-1|$ had to be when the Universe was a second old, or very young, we find out that the following inequality had to hold
$$|\Omega(t=1\,{\rm s}) -1 |\lt 10^{-62} $$
or so.
For some extended discussion what the flatness problem is and how the cosmic inflation solves it, see e.g.
http://motls.blogspot.com/2014/03/alan-guth-and-inflation.html?m=1
Now, if the Universe didn't have this fine-tuned $\Omega$ when it was one second old, the density would immediately begin to diverge away from $\Omega=1$ to either hopelessly high values (implying Big Crunch) or hopelessly low values (implying sparse space without stars) billions of years after the Big Bang. That's what the FRW and similar equations – effectively, Einstein's equations (the field equations from the general theory of relativity) specialized for a uniform Universe – imply.
Here everywhere, the cosmological principle is assumed, so the density $\Omega$ is assumed to be basically uniform across the Universe. Even if you insist that it remains a uniform, a tiny change – by "one star or much less" – would have led to consequences incompatible with life.
If you changed the average $\Omega$ by changing $\Omega$ "more dramatically" but only in a smaller region of the Universe (if you literally tried to remove a "seed" of what is a star today), the consequences for that region of the Universe would be even more dramatic – it would have crashed or diluted or became incompatible with life much earlier.
So indeed, the mass density one second after the Big Bang had to be adjusted with a much greater accuracy than "one missing star per visible Universe". I sort of vaguely understand why it may sound counterintuitive but there is really no paradox here. If the early Universe were a consequence of the present, then it would mean that someone had to adjust the number of stars etc. with the precision better than "1 star per Universe", and it would be extremely unlikely or unnatural if not a downright inconsistency.
But the point is that the events of the early Universe are not a consequence of the present. It's exactly the other way around. The present and the stars around (the "future") are a consequence of events in the past (the "past"), including the events in the early Universe. So it makes sense to ask what events or observations would be different if the early cosmology weren't this exactly flat – the answer is that (almost) no stars would exist 14 billion years after the Big Bang, It does not make sense to ask how the early Universe were different if the number of stars today were smaller by one – because the history of the early Universe in no way depends on the number of stars today (causality: future depends on the past, not the other way around). We don't have to fine-tune the number of stars in the visible Universe today with an unrealistic accuracy in order to allow the young Universe to exist. The young Universe did whatever it did long before we were here and we can no longer change that!
