Yeah, we're kind of at the point where we're to the question of what kind of complexity are they talking about here?
I think it's totally possible to have a class with few options, few builds, and few levers, and still be reasonably complex in terms of the mental load required to play one at the table. Barbs fall into that camp for me.
I've ruminated on this off and on today.
I think it comes down to a mixture of build, choice points, and rules knowledge.
A low complexity character doesn't really need much info beyond the basics. They don't cast magic, their abilities are active or reactive. Action surge is a use ability (you use it and then it's done). Everything else is basic rules 101 (attack, skills, etc). You can make a fighter or a rogue and have everything explained on their character sheet.
Barbarians require a little more tracking. Rage changes your combat stats and must be tracked to make sure it doesn't end. Further, rage fuels a lot of their secondary abilities now so a player needs to track rage far more than a fighter tracks action surge or second wind. I think they are low end average, but definitely more than a fighter.
Casters automatically bring in more complexity. The paladin and ranger add a little magic to the warrior chassis and the cleric and wizard are mostly spells with nothing else to track. Playing them well requires a lot of skill, but playing a blaster, healer, or smite/HM spammer is still effective.
Druid, bard and sorcerer are caster+. They not only are full casters, but have additional resources to track (wild shape, sorcery points and bardic inspiration) plus they have complex elements to their builds and interactions. While cleric and wizard both have abilities outside of spells, they are secondary to their magic. Wild shape, metamagic and bard dice are integral to their identity.
Warlocks have two things a new player needs to know: their magic doesn't work like ANY other caster in the game and it interacts with invocations in different ways. While it is possible to play a warlock as an eb spammer, you still have to be aware of the right abilities (agonizing blast) to take. Lastly, monks have a variety of things they can do, some cost focus and others don't, and they are always weighing various options.
Maybe a better way to think of it is: how much homework does a player need to do to understand what their character does. Low characters don't require much more than simple system understanding and their class features. Average has more complex features or spells, and high has very complicated features, spells or both. If you handed a player a pregen and the PHB, how easy it would be for the player to figure out how to play that pregen is what we're looking at.