It is correct that the set of possible $H_n$ over all the possible $S$ reduces as $n$ grows. However the attack evaluates the $H_i$ for a fixed random $S$, not for all $S$ simultaneously; thus that reduction is immaterial to the success of the attack.

When we model $H$ as a random function, then until there is a collision the $H_i$ are random, and hence the probability of collision among the $H_i$ with $0\le i<n$ is per the birthday bound.

It is correct that the set of possible $H_n$ over all the possible $S$ reduces as $n$ grows. However the attack evaluates the $H_i$ for a fixed random $S$, not for multiple $S$; thus that reduction is immaterial to the success of the attack.

In other words: it is evaluated the number of possible $H_i$ for random $S$ as a function of $i$, and from that drawn conclusions for collision probability among the $H_i$ for sequential $i$ and a particular $S$. The conclusions are thus not justified, and, it happens, quite incorrect.

When we model $F$ as a random function, then until there is a collision the $H_i$ are random, and hence the probability of collision among the $H_i$ with $0\le i<n$ is per the birthday bound.