Solocarry potential is a major problem
The ability to solocarry on any champion requires a tremendous amount of snowball and damage potential. And the amount of demand for the ability to solocarry is exactly why damage is so high. A very vocal portion of the playerbase demands the ability to solocarry, which in turn demands tons of damage and snowballing to be able to solo carry.
Because being able to solocarry means you have to have enough damage potential to burst down enemy squishies before they can do any serious damage to you. Be able to shred tanks regardless of their resistances. Push quickly enough that the enemy can't mount a strong defense against you.
Seem familiar? Towers are weak. So a Solocarry can have pushing power. Tanks feel weak. Solocarrying capability at work. Squishies die in a blip. Solocarrying! Because solocarrying means you need to be more impactful as 1 player than the enemy 5. And if tanks are tanky and towers are strong, there's much less solocarrying potential available.
Solocarrying potential in of itself is not inherently a problem as a mechanic, but the majority of champions absolutely should not be able to solo carry. Ever. Any spectrum of solocarrying potential for a champion that's not specifically designed to excel in a fight against multiple enemies (EG: Illaoi) is usually toxic.
A Yi/Yasuo/KaisaAny other DPS slaughtering an entire team is perfectly fine, IF said DPSer has an assisting team actively enabling it the whole way through. A Kaisa shredding a team from behind the protection of a Braum, with Sion and Swain leading in front as a deadly vanguard to prevent enemies from reaching her? No solocarrying there.
What's not fine, is when outliers begin to solocarry when not designed to excel against multiple enemies. Which is why the gold funneling strategies are bad for the game. It's built on pushing an outlier into an abusive state, it's not fun to play against. Should it be competitively viable? I'd personally want to say, yes, since it's cheese strategies that can be countered.
But realistically, no, it should never be viable. It's designed to leech player agency from the game, and funnel it all into one player so they can solocarry. It's not fun for most of the team that does it. It's a strategy designed on invalidating eight players so that one player will decide the game. (Sounds an awful lot like when a handful of marksmen snowballed ridiculously, doesn't it?)
It's not fun to play with for many. It's not fun to play against for many. It's not fun to watch for many. And it's a solocarrying strategy. A perfect example of why solocarrying when not designed to excel against multiple enemies is a problem.
And the champions that excel against multiple enemies have a common rule. They're immobile. Velkoz, who can obliterate an entire team in a rotation if they misposition, is slow, telegraphed, and squishy. Illaoi, telegraphed, slow, melee, easy to kite. Their solocarrying potential is situationally limited, and has plenty of counterplay.
What solocarrying in general lacks, is counterplay. Because it requires high enough damage to obliterate squishies. Shred tanks. Snowball out of control. Turrets have to be weak enough/have enough ways to take them down that it becomes extremely hard to stall out the game. Many of the current problems are symptoms of solocarry potential.
Note, I don't consider a player being able to exploit a messy and disorganized team as solocarrying, but rather, tactical play.