Checking whether close votes come from a single user
One of the question asked here is: "Do the mods have any way of checking whether the close votes seem to be largely the work of a single user?"
As far as I can tell, you're right in saying that regular users do not see who cast the first close vote until the post is actually closed. After the question is closed, the close voters are displayed together with the close reason to 3k+ users, and any user can see them in the timeline and the revision history. (A question might enter close votes review queue also after a flag to close rather than vote to close - but from the wording it seems that you actually saw close votes in this case. So you're probably mostly interested in the initial close vote.)
What regular users see is the number of close votes which have already been cast (shown to 3k+ users). Users who have 10k+ reputation can also recent history in the review queues, including the posts reviewed by others. (Any user can see the votes if they get the link to the review from somewhere - but 10k+ users have links to recent reviews available in the review history.)
Also Data Explorer (SEDE) can be used to see whether there is larger number of posts in the review queues. (Although one should keep in mind that the data in SEDE are updated only once a week.) Some examples of queries which might be useful in connection with this: Questions entered in the review queue (in the date range), Number of questions in a review queue per day, Number of questions in a review queue per month. (I will mention that the questions which have been deleted before the last update will not show among the results. But my impression is that questions aren't deleted on MO too quickly - perhaps with the exception of questions which are completely off-topic.)
Moderators can check on moderators' timeline of a post who cast the first close vote. Although this has to be done separately for each post, I suppose that if many closures were initiated by the same user, the pattern will be noticed after checking several posts. (And posts which need to be checked can be found through review queues or using SEDE.) I will quote here in full animuson's answer from: Can moderators see the close/reopen votes (and not the reviews)?
Yes, diamond moderators have access to a special timeline on posts which does list all close and reopen votes (and who cast them) inside it, including any that have expired or otherwise no longer count. Close and reopen votes are the only votes that they see in that timeline, though. This information proved valuable to me, as a moderator on Stack Overflow, countless times in determining what happened with a question, whether it be related to review or not.
Moderators do not have access to go to a user's profile and look at all the close and reopen votes a user has cast (because they don't have access to the votes tab on your profile).
Non-diamond users cannot see anything about close and reopen votes outside of the count that is provided in the link and the names provided in the revision history.
I am not a moderator on any site, so I cannot check this for myself. However, I consider the information in the linked post reliable. (Moderators are in better position to answer this question, but since some time has passed without any reply from a moderator, I have posted at least an answer with the information which I was able to find.)
Is that a problem?
You have also asked: "Do we have any safeguards against drive-by sessions or is it just an inevitable "occupational hazard" of using the site?"
Similarly as many other things, there is a daily limit on the number of close votes. According to The Complete Rate-Limiting Guide (current revision), the limit is "50 close/reopen votes/day/user on Stack Overflow, Mathematics, Server Fault, Super User and Ask Ubuntu, 24 on others".
Another thing which helps is that five users are needed to close a question. (With some exceptions, such as close votes by moderators and duplicate close votes using dupehammer by gold badge holders.) So perhaps the main issue is not whether somebody voted to close a question which really deserves closing rather whether they voted on several posts short after each other. (Of course, different users have different criteria on what is on-topic on MathOverflow and what should be closed.) A close vote pushes the post in the review queue where it is seen by other users who decide whether the post should be closed or left open. And if there aren't enough reviewers, the close vote will eventually age away.
So basically the only problem caused by this is that there is a larger number of posts in the review queues and more work for reviewers. However, on a healthy site which has enough users caring about site maintenance and community moderation, even if there is occasional increase of the posts to be reviewed, they will be eventually dealt with. (Whether or not the situation with community moderation and related issues on MathOverflow is good or not probably depends on your standards.)
I will also add that only 3k+ users can cast close votes. Hopefully, most users who reached this level should already have a good idea how the site works and what is on-topic/off-topic. So one can expect that they use the privilege to cast close votes responsibly. (One of the purposes of reputation is to achieve that various privileges are given to users who already have sufficient experience with the site.)