Is finding collisions in a part-hash not often enough a bad problem?
There are different birthday bounds when we draw independent uniform random integers less then $d$&nbsp; (for some large $d$, including $d=2^{32}$ of the question) and watch for collision(s):


In crypto, we often consider the bound of $\sqrt d$&nbsp; ($65536$ for $d=2^{32}$) draws, at which there is a fair probability of collision: $p\approx1-1/\sqrt e\approx39.3\%$. This is also about the number of draws for which it is most likely that the first collision will occur.
The number of draws starting at which it becomes probable ($p\ge1/2$) that there is at least a collision; that is about $\sqrt{\log4}\sqrt d$&nbsp; ($\approx1.17741\sqrt d$&nbsp;, $\approx77163$ for $d=2^{32}$); more precise formulas are here.
The average (equivalently: expected) number of draws at which the first collision appears; that is about $\sqrt{\pi/2}\sqrt d$&nbsp; ($\approx1.25331\sqrt d$&nbsp;, $\approx82137$ for $d=2^{32}$); more precise formulas are here.


The difference between the three is because mode, median and mean of a discrete probability distribution are not the same thing (sort of: we rarely have a collision much before $77163$, and often have the first collision way after that, hence the significantly larger mean). The confusion is probably what causes the discrepancy in the question.


Image credit: Cmglee; source; Creative Commons license.



If for some would-be hash function we consistently observe collisions much less frequently than predicted by these bounds, for would-be random input (restricted to a domain still much larger than the output domain is), then


our would-be random input (or its domain) is biased
and our hash function is not secure in the random oracle model, in a manner that is related to the bias in our random input.