I’ve never seen a poisek officially pasken that it IS permissible for them to vote (perhaps it exists though), whereas here you see Rav Kook officially paskened it is not. Furthermore, even where a Rov will tell you of course women should vote today, it is clearly a b’dieved. Since general society already allows it, if we refused it we would lose half our influence. Therefore, absolutely women today should vote since it is legally available and the secular, irreligious etc. will outvote them if we shield half our eligible voters.
Nevertheless, IF given the possibility to discontinue (or better not allow in the first place) woman’s suffrage across the board, I highly doubt any Godol would disagree with Rav Kook on this issue. I challenge anyone to find any Godol on the record who ever stated that we are better off with this suffrage. It simply does not exist.
Whether the reason for this is black and white halacha, societal, detrimental to families, etc. is almost secondary. Most situations are not black and white. That’s why we have Gedolim and poiskem to help us sort out these things. Even if we disagree with them, using our own krum logic (which we stupidly think is impeccable logic), we follow our religious leaders.
We are all, unfortunately, living today in a world of b’dieved’dik matzif’s.