I think the problem may depend on browser rather than OS. I am on a PC right now, and left-clicking the YouTube popup link works perfectly on Safari, but not in Chrome, Firefox, IE, or Opera.
However, I get this type of script error in Inspector about 15-20 times when viewing the site in Chrome.
Uncaught SecurityError: Blocked a frame with origin "http://www.mildenhire.com" from accessing a frame with origin "https://apis.google.com". The frame requesting access has a protocol of "http", the frame being accessed has a protocol of "https". Protocols must match.
Interestingly, I can still right click on your link and open it in a new window, only left-clicking does not work. Perhaps these errors are somehow blocking the javascript YouTube popup? I hope this can at least point you in the right direction.
UPDATE: Just to confirm, left-clicking on the button is what's causing the error above, registering 1 error per click. However, it might be safe to ignore.
In Firefox, clicking the button returns this:
Error: Permission denied to access property 'nodeType'
The common denominator I see with both error messages is line 13 of plugins.js, however that line is incredibly long and I can't find anything meaningful in it (JS is not my strongest area, to be honest).
But, since no one else has answered yet, what I would try is temporarily disabling the script that includes line 539 below, which contains the origin "https://apis.google.com" and see if the popup begins working correctly.
po.src = 'https://apis.google.com/js/platform.js';
I may be far off in reasoning that this is the problem, so I encourage someone who knows JavaScript better to point us in the right direction. However, I do have a hunch that it is connected to the pop up script being "blocked."
Otherwise -- I hate to suggest this, but since you're desperate -- would it be possible to use a plain html or non-js/jQuery method to open the YouTube window? Good luck!