What reasons for anonymity of voting overcome the following objection?
A down-vote fails to communicate what is thought to be wrong. Suppose I post a question that says "Doesn't the fact that $2+3=5$ mean that $2\cdot3$ must also be $5$?". Someone down-votes it. It may be that they down-voted it because they thought I was mistaken in thinking that addition is the same thing as multiplication. But I may have had doubts about whether the number $3$ exists and I may suspect it was down-voted because of my confused and erroneous belief that that number exists. Or I may mistakenly think it was down-voted because it's not research-level math, despite the fact that most mathematicians know that the proposition that $2\cdot3=5$ is one of the great unproved conjectures. Or I may suspect someone down-voted it because he thought I'm having an affair with his wife. A simple verbal statement saying "It is well known, having been proved by Paul Erdős in 1939, that multiplication is not the same thing as addition." would make it clear what the objection is. A non-anonymous downvote would make it possible for me to ask the person specifically what the objection is. A system that communicates the fact that there is an objection but withholds all information about what it is cannot indicate in what direction content should be altered to make it appropriate.
Your objection is indeed a reasonable objection to anonymous downvoting.
Let me list several reasons in favour of anonymous voting, without attempting to evaluate whether they 'overcome' it.
It is how the software works, it seems extremely unlikely that the SE folks would implement public voting across SE, and rather unlikely that they'd implement it for us.
It allows rubbish questions to be removed quickly without giving a troll, spammer, or otherwise unwelcome poster particular people to pick fights with.
It is important to the site that good questions receive high vote counts, and bad questions receive high negative vote counts. (Greater spreads convey more information, and provide stronger feedback for posters.) Non-anonymous voting would, in my opinion, severely restrict the overall voting rate.
It ensures that people feel able to reward (or 'punish') questions which they like, for reasons particular to them. Public voting might result in all votes being held to a 'standard', such as "I have read the answers and certify it's correctness and relevance" which would be way too high for the site to function in anything close to its current form.