I agree--and seems like so do most people--The Eagles belong to something different than yacht rock. Though Eagles bassist Timothy Schmidt can be found moonlighting backup vocals on some Toto records and Jeff played a number of Don Henley hits and they knew each other well, etc......they're different.
The documentary, iirc, pointed something out that I agree with, and I don't think it was the first to say it. What typically gets called "yacht rock" has an underlying R&B influence. It's not R&B, but it was a time and a group of people who were listening to R&B and importing elements of it into "white boy music."
This applies to chord progressions, instrument techniques, and all that stuff, but it also applies to the basic sound of those albums that I personally think define their ethos. The very engineering, the mixing and the mastering, the production decisions, sound more like the R&B stuff back then that I'm familiar with than the pop-rock or So-Cal country rock of the time. Take Jeff, probably the most ubiquitous yacht rock drummer in the scene....he was fascinated with those old Detroit-based Motown drummers, and he started using their dampening effects on his drums, tuning like they did, all kinds of little things that subtly change the drum sound. And then the engineers would wash out certain EQs and boost others that were mimicking some of the same things. Similar things apply on other instruments.
So that brings me to one of the major ways I tend to classify yacht rock. The actual sound of the album, as much as or more than the songs and their structure. They sound a certain way. Sure, both The Eagles and Toto of the 70's have a unifying 70's sound to their albums indicative of recording techniques and technology of that time. But they can then be pretty reliably separated even further, I think, into the regular rock or pop stuff that had been evolving, and the stuff that was borrowing heavily from what had been going on with "the other side of the radio" as Jeff called it, referring to stations that played predominantly R&B.
Then beyond that, you've got the piano/synth/moog sound that drove a lot of yacht rock. The Eagles probably have some piano here and there, but their music was largely guitar-driven. A lot of yacht rock was driven as much by electric pianos as by guitars. But there again, that's a clear R&B influence on the way a lot of those songs were written around piano instead of guitar, and everything that goes with that.
So that brings me to "Africa." Here I will respectfully disagree and opine that I think it comfortably belongs in the yacht rock category. I'll try to explain why it feels like yacht rock to me instead of a rock anthem, and while I can't speak for others, I suspect they're thinking something similar.
One, because it has the yacht rock sound. A very different song than, say, "Afraid of Love" off the same album, but tonally, the entire sonic ethos of the instruments still signal the same album, the same time. And again with "Waiting For Your Love" from the same album.....more or less a straight-ahead R&B track, but everything about the sound screams to me the same things all the other Toto IV songs scream. Or, jump to a totally different album. There are things going on with the sound of IV that are also going on with something like Michael McDonald's "If That's What It Takes" album. If you skip ahead to 1990, McDonald put out an album called Take It To Heart, and as an illustration of distinctive sounds, many, many years later I came across a song of his I'd never heard, but everything about it screamed "This has to come from the Take It To Heart sessions, surely." It sounded just like the other songs, in a way. A few years after that I got to ask him myself about that song and he confirmed it was recorded for that album, and something happened....record company decided not to use it or something....can't remember exactly what he said. The larger point being that, because I conceptualize yacht rock at least partly in terms of the sound and independent of chord structure or playing style, Africa just sounds like yacht rock to me.
Second, because of the other things too, like the actual style of the song. I'd never thought about it being a rock anthem until you said that. Now that I think about it, it's certainly anthemic, particularly the chorus. I still don't see it as a very rock oriented tune, though, because the defining and driving elements of it are keys and synths, not guitar. Most of the track doesn't even have electric guitar, just a very sparse acoustic line. An electric line does come in on the third chorus and it definitely adds a lot and is even iconic to some, but it's still not driving the song. It's decorative. A background part, if you will. For me, a rock anthem is going to be driven and centered around the guitar. Which is why I'd call a song like Hold The Line a rock anthem, because even though it begins with an iconic piano line, it's the guitar kicks in and becomes the main driver of the song. Not to mention a great guitar solo midway through. Africa, of course, has no guitar solo, and not too much guitar overall. There's also a metric ton of percussion in Africa, which characterized the yacht rock ethos as well. Certainly rock songs have used percussion, no doubt about that, but they don't tend to be saturated in it. And then there's the drum loop. For me, a rock song will typically feature a drummer doing his thing all the way through. Famously, Jeff looped two measures of his actual playing into one of the first-ever pre-digital loops. He wanted it to sound like the Lin drum machine lines that were popular at the time. Something that would be repetitive and more static, but that felt better than a drum machine, so he played it himself and picked out his favorite to measures. Then, of course, he had to overdub the drum fills leading into the choruses of Africa, but he wasn't actually playing that song. The intention to have a loop-ish, drum machine feel isn't necessarily a yacht rock thing to do, but it's an R&B thing to do which goes back to the earlier points, and it's definitely not a very rock thing to do.
The weird thing about Hold The Line is that as rocky as it is, it still has the yacht rock sound, and I wouldn't necessarily try to take it out of that bin, though I'd understand if somebody else did.
So, that's as best as I can explain it. Your mileage varies, and that's okay by me. Songs hit you how they hit you. This is an attempt to explain why I suspect people think of Africa as a yacht rock song.