So this means that just because that video you've inserted from Google Drive into a Google Site, may play beautifully on your Mac/PC, but that does not mean your iPad will play it.
Simple solution? Anything on YouTube is iPad friendly, so uploading your videos to YouTube (or Vimeo) and then inserting them into your Google Site will work, on any device; and as we're an educational domain, no dodgy mosaic or ads, unless you are using Blogger, in which case, you will see a 'video mosaic' at the end.
Like this one:
So you could have two videos, one inserted into a Google Site, and one from YouTube, and the first one will not play, even though it is the exact same video, before it was uploaded (and converted) it to YouTube.
To see this phenomenon demonstrated on a Google Site, click here.
The good news is that sharing a video straight to your YouTube account straight from QuickTime is very easy, and will do all the converting for you: