Well, let's look at well known melee carries in other games:
Dota 2
Sven
Why is he viable? One of the highest late game DPS in the game that is being applied AoE and has a ranged AoE stun. Plus provides a decent defensive steroid for his team. Due to how Strength works in Dota, his damage scales with items that also give him health so he becomes tankier as he deals more damage. While he is melee, his cleave effect can make him have some form of poke in lane as long as he can attack someone from the right angle.
Juggernaut
Basically Master Yi on steroids. He has multiple ways of becoming immune or untargetable to all or certain forms of damage, has a CC Immune effect, and also provides insane team healing through a decent duration, mobile ward that grants healing based on % max health. He also has crits, which are much more restricted in this game (something to think about as well).
Slark
A super slippery fish that has an ability that makes him untargetable by all means for a flat duration and also a passive that grants him absurd health regeneration while he is not seen (and tons of movement speed). He comes packed with an amazing CC cleanse and a situational crowd control that makes escaping him decently hard. Finally, his E allows him to steal stats from his opponent and convert them into bonus damage and defenses against physical damage with his basic attacks.
Heroes of the Storm
Illidan
Stupidly mobile and packed with tons of sustain, he is incredibly hard to 1v1 without mechanics that counter him or intense crowd control and damage. His E allows him to dodge basic attacks very often and both of his R abilities grant him even MORE mobility. He is basically a form of Jax if Jax's mobility was thrown into orbit.
Samuro
(Jesus christ this guy...) Samuro is VERY insane for a character. He can become invisible without fade time and that invisibility cannot be broken by damage for the first second of it. Not only that, but he comes with innate crits (which are ALSO a more restricted mechanic in this game) and can spawn clones of himself to confuse his opponent while also offering him infinitely more possibilities through more damage, split pushing, or just being meat shields.
So, after looking at five rather successful examples in other game (when they are balanced correctly), what are some common occurrences? Wel...
- Most have some way to dodge or negate crowd control effects. Those that don't usually have something that they have access to later in the game to negate crowd control effects.
- Most provide team utility besides just flat out shredding enemies. This can be through crowd control, meat shields, or support-like utility. Most of this utility also boosts the effectiveness of the melee carries themselves as well.
- Most have a way to deal serious AoE damage. Their DPS isn't usually restricted to one target and if it is, it is absolutely insane.
So what makes a successful melee carry? Someone who can deal good sustained damage to multiple characters at the same time while also being able to combat the mechanics in the game that would stop them from doing so (crowd control). Even if behind, these characters can provide fairly decent utility to their team that isn't incredibly powerful overall, but often very powerful in certain situations. Most are also fairly durable or have mechanics to help them negate damage.
The champions that currently provide these in our game are:
The ones that have failed so far are therefore:

HOWEVER, are the Melee Carries in LoL currently designed to be healthy and fun to play against? No. Not for the style of game that LoL is. Yasuo, Fiora, Riven, and to an extent Ekko are also very annoying to play against and why? They have mechanics that make immune to damage or punishes people for damaging them! Or you are Riven and have gameplay bugs that have become a part of your character. Anyways, LoL's successful melee characters are poorly designed because they often punish most forms of retaliation towards them! None of the characters I listed above can negate damage as often as the melee carries in LoL and, if they can, it often requires certain requirements to be met or costs them DPS. Most of the successful melee carries in LoL do not suffer the same rules and therefore can sometimes just feel unfair to play against.