Aah, but you’ve opened a can of worms, haven’t you?
In a subtle way, you’ve touched upon the notion of ‘natural gender’ (as opposed to ‘grammatical gender’, per se). Yes, it’s true – rocks are neither male nor female; any notion of ‘gender’ that is assigned to a rock is purely a grammatical construct. (If you want to call your pet rock “Ma’am”, that’s up to you. ;))
But, there’s the question of ‘natural gender’ and how it relates to grammatical gender; and this notion varies by language. In general, nouns in Koine Greek don’t follow natural gender (that is, ‘child’ doesn’t change gender based on the gender of the child being referenced) but pronouns do.
So, how do we approach the question of ‘spirit’? Does the use of the neuter with Πνεῦμα indicate simple gender agreement (i.e., neuter nouns take neuter modifiers), or does it indicate natural gender (i.e., the Holy Spirit is neither male nor female, so neuter is the appropriate natural gender)?
Of course, the OP’s original question only touches upon this consideration. In particular, it seems that he’s noting the (American English-language) tendency to utilize ‘he’ or ‘she’ to refer to persons, and the aversion to using ‘it’ to refer to persons, and asking whether the same rule applies in Koine Greek. (Think about it: does it sound jarring to say, “the baby was tired, so I put it to bed”? In general, that would be proper (unless we both know the particular baby and its gender, in which case we might say “I put her to bed”), but it just ‘sounds’ wrong in contemporary American English.) So, generally speaking, given our aversion to use ‘it’ to refer to a person, one might conclude that using ‘it’ tends to imply inanimacy!
The question boils down to the following: does the use of neuter pronouns for ‘spirit’ in Koine Greek indicate a lack of personhood (as it might in contemporary American English)?
So, we’ve come full circle, it seems: Koine Greek is highly inflected, so we expect grammatical rules to dictate the gender of nouns and their modifiers. However, the notion of ‘natural gender’ also exists, tending to result in pronouns (not nouns!) indicating the gender of persons; in these cases, the gender of the pronoun does indeed “indicate the gender of the word’s subject”! Nevertheless, the assignment of neuter gender in Koine Greek (i.e., ‘it’ as opposed to ‘he’ or ‘she’) would seem not to make a statement about personhood in the way that it would in contemporary American English. That’s a hang-up that we seem to have developed (God bless our sensitive souls, desiring compassionate gender-neutral language!), but that wasn’t a hang-up of 1st and 2nd century Koine Greek-speakers!
(Of course, none of this discussion changes the answer to the question that’s puzzling the OP: “No, the use of ‘it’ in Koine Greek does not indicate that the Holy Spirit is a non-person”!)