AJAX calls should absolutely work, especially since it looks like your site is actually on a Plesk environment. I don't think it's a block on the file, but your screenshot indicates a 500 ISE error which can be very generic. Because of that we could assume almost anything; like It could be the call timing out, using too much memory, or etc.
Try turning on wp-debug, and consider reviewing the php settings via a phpinfo.php file. Even on a Plesk platform you should be able to modify the settings if needed via a .user.ini file.
A few other items to consider:
Are your perms correct?
What if you disable plugins?
Switch to the default theme?
Could be bad transient data?
Hopefully this helps out. Let us know if you manage to work it out on your end.
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