Sorry, we very rarely give flat-out rejections like that in Support articles, but this is one of the few aspects of Finalsite where this is a hard-and-fast rule. It's actually not even a Finalsite thing, it's more of a fundamental consequence of the nature of emails.
While it is technically possible to find various ways to include video players in emails, you really, really shouldn't. Email doesn't work with video for a couple of reasons:
1. Lousy media support in email clients. A webpage with video embeds will look pretty much the same in Chrome, Edge, Firefox, Safari, or any other desktop browser. The same is not true for a message displayed in an email client (like Outlook or Apple Mail or GMail); there are basically no good ways to make sure that an email with any kind of video content will appear the same way from one email client to another. Many email clients suppress videos entirely because they're so closely associated with email spam (see below). Others will hack up the embedded media player to the point where it won't work, if they show one at all. Many users will just see a blank box in the middle of your email. So you don't get to choose how it appears, and you have no guarantee that your content will even show up.
2. HUGE data push. Embedding an video in an email would mean sending a copy of that video file out to everybody on the mailing list, pushing a massive quantity of data out over your network when you send the email blast. It also forces everybody who checks their email on their phone to pull down a gargantuan, un-optimized video file, possibly over expensive and slow cellphone data connections.
3. It's shady. Embedded videos are a hallmark of spammers and will negatively affect your sender reputation. That could lead to your emails being blacklisted by other email servers (you never know what kind of a hair-trigger other sysadmins may be on when setting up their security parameters).
There are other reasons, but those are the biggest.
It's much more elegant to include a link that the user can click which will detect their device identity and serve the user a video file that's sized for their screen (rather than one that may be way too big. The only way to do that with an embedded video would be to embed ALL of the optimized versions of the video, which would mean 4 or 5 video attachments of various sizes on every email sent!)
Linking to a video rather than embedding it also allows end users to decide whether or not they want to download the media content in the first place - if the video is embedded in the email, you remove that choice (and seriously aggravate a huge chunk of them. At least one person will go ballistic at not having a choice about downloading a video file over a cell phone data connection.) And embedding a video doesn't even save a click or a tap - users would still have to click or tap on an embed to get it to start playing, so you might as well use a link to an optimized web stream instead.
Virtually every time you see what looks like video in an email, what you actually see is a still image of the first frame of the video, with a triangular "Play" button photoshopped on top of it to make it LOOK like an embedded video. The image is linked to a webpage that serves up a video file through a regular media player. This is by far the most practical (and polite!) way to distribute video content via email. Every media object in Multimedia Manager has its own URL.
If you link to that direct media URL from an email, users will see an optimized video player in their browser.
Again, we usually prefer to give options for how to do something rather than advise against it. But we've been down this road a bunch of times and it never ends well. Is it technically possible? Sort of. You can force a video into an email, but not all of your recipients will see it and there's no good way to control what happens if/when it's stripped out - so the only thing you know for sure is that some people will see some thing that you don't control, which is probably the least-good outcome for your emails!
The alternative of using a linked still image to deliver an optimal video when the user requests it (and not before), via a method that is supported by all email clients, means that your emails remain consistent for every reader and you maintain control over their experience with your school communications.