Unfortunately, your demo does not work for me. But I have a similar demo that we exchange with Clients in an older forum post here Handsontable example - JSFiddle - Code Playground maybe that would help. In general, that is a custom code so there’s no guarantee it will work in all conditions. I tested it on Chrome 120/mac OS Ventura.
The problem for me is that the tooltips appear several rows below and a few columns to the right of the header. And, they are in different locations depending on the screen size or orientation. If I adjust for a full width browser like this, they are in completely different locations on other browser widths.
I’ll do my best to help you with debugging if you could share the recent progress of this task in a demo and specify which browsers do you need to support.
I tested the demo and got out log from the afterOnCellMouseOut hook so surely the hook logic runs. I tried to show tooltips with 2290 px - wide window(saw all columns and some empty space) for the JSFiddle demo, and they did not appear so I do not think that the screen size is the issue.