Well you could try the easier test first, and if it proves to be the fix, it will not require so much trial and error. That would be to (on a copy of your page, save what you have) revert to exactly the hover tooltip code on the w3 schools site that works. Don't use any of the modifications that you've added. If that fixes it, it's not a conflict with your basic styles, rather just a modification to the tooltip code. Then you can add those few things back one at a time until it breaks. Then, if it's something you need, but don't know how to use it there without messing up those browsers, ask us, we might have an idea or two.
Even if you it appears to be a conflict with your site's style, that's easier to test than you might think. You can remove whole blocks of style at a time to find the section you need to concentrate on. But try the simpler part first, it may provide the answer quickly.
Oh and I agree that you might as well leave the tooltips in regardless, because most people will still see them.
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