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The page which doesn't work has the sandbox attribute in the video IFRAME element's properties, the page where the video works does not have that attribute. This is all about content authoring which is unrelated to Admin Tools. It's something you –or whatever YouTube embed method you're using– did. Note that using the sandbox attribute without any content would only allow plain HTML and CSS to load. That's okay for text pages, but it will of course not work with videos which need JavaScript to provide playback controls, start a full screen display etc.

I would also like to note that the browser's Console (in dev tools; hit F12 on your keyboard) was very clear that the problem is a sandboxed IFRAME. You need to consult it before making assumptions about what's wrong.

Further to that, you could have already deduced it cannot be an Admin Tools issue without even looking at the console for a very simple fact. If Admin Tools is blocking something it is blocking it across your entire site. Having one page working and one page not working is an immediate indicator that there is something different on the page which is not working. That's what led me to look into the browser's console . I then simply compared the video IFRAMEs between the two pages to verify it's the sandbox property added on the non-functional video's IFRAME.

Finally, I'd like to note a terminology confusion on your part. You do not have any "Java" issues on your site, as you are neither using any Java applets, nor does your site run on Java (it's Joomla which runs on PHP). You have a JavaScript issue. The difference between Java and JavaScript is about the same as that between car and carpet. Please do use the correct terms. Had you not mentioned "video" in your issue I would have been unable to help as you'd be asking me for help with an issue you not only do not but can't even have on your site.