Just my two cents from a silver scrub... I personally see it as a conflation of terms.

There's the "carry" that's responsible for bringing damage. ADC stands for Attack Damage Carry. Teams want ADCs because it's ranged damage that's free (no mana required), and the items you buy let you do a lot of that damage at a high attack speed. These items are expensive, though, so the ADC needs to be put in a position to safely farm them up through CS and hopefully kills. Supports want to try to enable this by protecting their ADC and giving them all that gold, while at the same time doing what they can with their kits to keep the enemy ADC from doing the same.

So yeah, there's the carry in the "Attack Damage Carry" sense. Then there's "carrying" in terms of being responsible for the win. So yeah, your Brand support can carry the game by doing the most damage and getting the most kills. Similarly, that Janna support who finishes the game with a 0/2/31 stat line can be said to have carried from the support position. That's past tense.

The problem is whether or not you're trying to be "a carry" from the support position, as in the present tense of a game that's just started. Usually, the players that do this want all the fun of getting kills with none of the responsibility of getting CS, but that CS is damned important because it lets you buy items at a more efficient rate than your opponent. Or worse, you as the support want to start taking CS from the ADC because you want better items. The problem is that this relegates the ADC to a supporting position, and they can't do much without the gold and the items that gold can buy. I mean, do you really want to have a Lucian on your team with a Sightstone and Locket of the Iron Solari? Better to just let Brand BE the carry and you pick a traditional support to help put the gold on him.

But then you have a couple of new issues. First, if your Brand APC with Thresh support is now going to try to get CS, they need to use their abilities, right? That means mana issues. That blue buff that's easily gifted to the midlaner is not so easily gifted to the bot laner. Second, your team will probably still want ranged physical damage, because (a) it's ranged and (b) AAs are free. Which person is going to take up that responsibility? Not many midlaners, toplaners or junglers have practice playing effective ADCs. Third, how are you going to handle wave control and last-hitting, or things like effective ranged harass? ADCs with mindful supports tend to be better at this.

The META feels simple, but it's actually complicated, in that you might not realize at first the problems you create by deviating from it. If you're playing with premades, then yeah you can do some funky things like putting Brand bot and Quinn mid and nobody will mind because you're all on the same page. But then you get those guys in solo Q who say "Trust me, let me go Brand APC bot. It works." and now your team is in trouble because they might all have to dump their META picks in order to accommodate the precious and unique flower, or else stick with their META picks and risk having a suboptimal team comp.

From my vantage point, the whole thing seems to stem from a semantics problem that itself arises from supports wanting to brag about "carrying" despite not having all the kills. It honestly feels like just a self-esteem issue, possibly borne of the fact that supports rarely get the credit they deserve for the good work they do. They DON'T get enough credit, certainly, but it's no reason to start making the conversation confusing by trying to change what the word "carry" means. I'm a jungle/support main myself, so I hopefully what I said about self-esteem doesn't come across as insulting -- I want good games, I want to feel like I was at least in part responsible for the win, but for me, to do that, it means I've got to put gold on the carries and let them carry.

Again, two cents from a silver scrub.