(This is a long response, so I've had to split it into two posts — )
Part 1:
First of all — your lawyer is into it for money! Of course he's going to tell you to jump through hoops.
Second, as far as I can tell, emoji have not acquired intellectual property rights. Any emoji you *buy* may be the same infringement on Apple's (Color) Emoji... which seems to be somewhat original depending on how close it comes (and why else would you be using it?)
Third, emoji are implemented as part of the Unicode Character set (that is: "type") and there are arguments as to whether emoji are an emerging set of language characters — having meaning, in which case, that enforces the case for "type", characters, letterforms. I have not heard any news about any company going after those rights (they'd be stupid.) Emoji are too much a part of the current culture worldwide — and if something is in that "common use", it's hard to scoop a copyright on it. (The United States Supreme Court has shut down most efforts to copyright things that are considered "common use".)
As far as I know (and this goes back a l-o-o-o-o-o-ng way) Apple has never gone after anyone who legally obtains their software for copyright infringement. They had a pretty nasty scuffle with Microsoft awhile back (and deservedly so! I read the Windows programming manual — it was almost word for word the same as Inside Macintosh! They still lost.)
Furthermore, Apple provides sufficient user materials for you to make use of the Apple brand which reinforces their marketing standing. There was a time when you could not see a local attorney's television advertising without hearing Apple Loops in the background — no credit in the ad, BTW. (I'd watch for it) — and that's from attorneys.
Finally, Apple doesn't really make any money from emoji, per se. They make their money selling hardware (iPhones, etc.) that utilize emoji (they were the first to include an emoji keyboard for texting.) Apple does not sell Emoji Character sets. You are not selling their emoji (of course, they're not giving it away either...). There is "no harm" (that I can see) if you use Apple's Emoji in your videos. "No harm" is a pretty big deal in copyright disputes.
One final way to determine whether or not you will have an issue: Publish the video. If there's a copyright problem, YouTube and some of these other services will tell you (and give you a chance to take it down, or give up your monetization of the video... been there... done that!) One thing I would recommend though — if you have misapprehension about this: PUT THE CREDIT FOR THE USE IN YOUR VIDEO: Emoji Character images/set provided by Apple, Inc. or something like that. That does more for Apple than them squabbling over your use for the emojis they put in your hands for free. You should credit everything you haven't personally created *anyway*! (But if you're consulting lawyers... you already know that.) You cannot credit anything OUTSIDE your video, as in comments, About or Description sections... it doesn't count. OH! And put your © notice IN the video as well. Credits state your right to use the content you used and the copyright takes any problems the hosting services might have for their liability away from them. Have guts.
Now, if you were creating software for any kind of distribution in which you were going to place the Apple Color Emoji font... then you'd be in some trouble. I wanted to use the San Francisco Display font for a couple of FCPX Templates I created, but the developer's licensing agreement made it pretty clear I couldn't. It's basically the same thing: San Francisco is provided free with the Mac OS (but Apple made that particular font an OS only System font.) And, whereas I was able to find this information for San Francisco — I still cannot find anything similar for Apple Color Emoji font. Using the font that way (to be included with something else) is a whole other kettle of fish.
One more thing: I've been in several (not a lot) copyright disputes over the years. Almost always, the "offended" party gives you an opportunity to "Cease and Desist" before taking any further legal action. In my experience, it has usually been rather cordial as well. (US Olympic Committee asked me to cease and desist on my FREE Pyeong Chang 2018 Olympics font which I considered "fandom" and they saw it as a possible hit on their promotional items income. — Okay, I didn't think of it from that perspective at the time. I immediately apologized — I hadn't intended any harm to their business — and complied and that was all there was to it.) I still use it as an informational example here: https://fcpxtemplates.com/about-fonts-made-for-3d/ but the font itself is not available to anyone anymore (a few people got it before I pulled it).
... end of part 1