$S'_1$ indeed can have many extra gaps over $S_1$, but it doesn't matter and the distance $d(1,j)$ between the final strings in the MA: $S'_1$ and $S'_j$ would be exactly the same as $D(S_1, S_j)$.
Why? when we start the $j$ iteration, we do an optimal global alignment between $S'_1$ and $S_j$. At first, it looks as if $S'_1$ might contain disturbing gaps, which will cause the distance to get larger than the distance to $S_1$, and helpful gaps which will cause the distance to get smaller than the distance to $S_1$. Let's show that both aren't really helpful or disturbing! suppose
S1 = abc:
Disturbing gap - e.g.
S'1 = ab_c and
Sj = abc. as $\delta(-,-)=0$, whenever we encounter a disturbing gap at $S'_1$ we can just add an extra space to $S_j$ at no cost and get the same distance.
Helpful gap - e.g.
S'1 = ab_c and
Sj = abxc. It isn't really helpful, as if the global alignment would have wanted to add a gap to $S_1$, it would just add it - it makes no difference that that gap was there before we started the alignment...
Gaps added after the $j$ iteration - if the center star algorithm will add gaps to $S'_1$ in iteration $k > j$, it will always adjust strings from previous iterations with gaps at the same place, including $S_j$ - and again because $\delta(-,-)=0$ it doesn't make any difference.
To summarize - as there's no help or disturbance, the alignment distance would be exactly as the global alignment $D(S_1, S_j)$ would have wanted it, plus some extra aligned gaps which can not disturb our distance.