I have resolved the issue – but please read what I found as it may be useful.
Performing the steps you’ve requested the console shows one warning when the page loads, but clicking the close button literally does nothing – no additional activity in the console. The warning on page load is as follows:
DevTools failed to parse SourceMap: https://[redacted-domain]/wp-content/plugins/activity-log-mainwp/assets/js/dist/index.js.map
This warning doesn’t appear in the console during any other navigation around my MainWP dashboard. I’m using the default WordPress theme Twenty Nineteen. The only plugins I have installed are MainWP extensions and UpdraftPlus Pro.
HOWEVER: I was using Chrome. In order to provide as much information as possible, I also decided to run the same process using Firefox. However, when I used Firefox, there were no warnings and the nag did close successfully. Of course, now that the flag has been set, the nag is also now gone from my Chrome instance. I’m satisfied with that.
To be clear, I observed this behavior using the latest version of Chrome on 2 separate Windows 10 computers – so I know it’s not MY specific computer that has an issue. It’s definitely something in Chrome.
And, yes – you call it an advertisement, I call it a nag ?? Great product by the way – It’s on my list of plugins to buy once I have a few more clients on the books.