I believe the video picks on white mage because we've always picked on white mage. And it's likely to add to the mean-spirited chuckle because that seems to be his gimmick.
But let's consider the why of what he said. I believe the why is that the game has a wide variety of player levels and interests and having an "easy" healer is good for players that want a fairly reliable experience in content they're comfortable with. Realistically, white mage is a great choice for that. It's got the brand recognition that comes with FF and a reputation for being The Healer. I know it's hard to understand but some people feel like they're adequately challenged and enjoy the pace and they don't WANT more. In order to cater to a wide base you do need one version of each role that has a low barrier of entry and maintains a casual level of engagement. I would still argue that we could still have an improved base system with something to at least master or options to exceed that aren't necessary to succeed, but any more than that and I'd just be describing a whole rebuild of healer from the ground up (which I know most people here want, but i think that ship has sailed.)
As someone who prefers to play SCH or AST, if they decided to make one of them the simple one, I'd take the lump and figure out which I actually like playing. But we can't pretend that there isn't a need for an "easy" healer.
That stated, it would still be bad to leave WHM, whether we consider it entry-level healing or not, as-is because the fundamental problem is that occasional ogcd heal and ruin ruin ruin is not engaging in the slightest, and it leaves WHM without an identity (unless we consider Glare Mage an identity, which i don't.)
The problem overall is that there's an idea that every single healer needs to be equally accessible, which simply isn't true. The fact is we're a limiting role that needs some draw in order to keep people filling out parties, so if anything we should be making each even MORE distinct. Otherwise why would more than one type of person ever choose to heal? Limiting a role that is integral to the fights to one type of gameplay is bad for queue times on top of being bland and uninteresting to a wide swath of people you could be encouraging to heal. Maybe White Mage should be an entry-level healer for people who are not sure about healing, but it shouldn't be hobbled by that identity.
(There needs to be an unintimidating tank and dps as well, but that's not really a problem right now considering Summoner is mashing whatever button is highlighted on their hotbar and getting top dps, and Warrior needs no one. You'd think it'd be Paladin, but as ever they're just treated as warrior with softened edges.)
TL;DR I think largely the video does a good job of describing the problem and the rest is for comic effect. sorry for the essay