I think the issue of bot lane is not about income. Here's some reasons why melee is not great bottom:
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Melee typically have higher base stats, they want levels early on. Being a lvl behind where they would be in a solo lane is pretty detrimental for them.
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Objectives are easier with ADC. A lot of ADC have functions that let them do more damage to towers like Jinx, Trist, Vayne, Lucian, even Ezreal. The main thing is simply their range makes it safer for them to go up and hit a tower.
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Scaling is a big deal. Now while something like a Fiora or GP scales very well into the late game, the fact that they are melee and spell reliant makes it harder for them to carry a team fight. They have to go in and out of the fight, play around cooldowns, try not to get cc'd, try not to die to ranged carries. While an ADC can potentially stand at max range and just consistently auto down one person at a time. Now obviously melee carries get penta kills or carry team fights from time to time, my point is it's just a little easier with an ADC.
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Really susceptible to poke. Certain things like a Karma who can be very aggressive or defensive can make life hell for double melee, pair that up with an ADC and getting levels could be torcher. Yes, melee bottom laners probably have a very strong like lvl 3 all in spike, probably again at 6. But getting to that point with enough health and farm to actually secure the kill can be tough into a support + ADC combo.
I mean theoretically anything can work anywhere right? It's just a matter of ease. Putting a melee carry bottom is just really risky. Too difficult to pull off to make it worthwhile. So here's why we stick the ADC specifically bottom:
While all champs scale with level to some degree, ADC more than any other class need gold. They actually don't mind being 1-2 level behind solo laners if they get their gold because they auto attack, meaning the base damages from spells doesn't matter much. They just want AD, Crit, Attack speed so they can wack you over and over. Other classes are more suited to using their spells and base damages early on to create plays and advantages, so you want them in solo lanes. It's a win win because you put the ADC, who needs money but not a xp advantage in a lane with a support, who doesn't need either. So he shares XP but doesn't mind it since his partner secures farm for him. He has a partner to help him make sure he gets his money, they protect the flimsier bot tower with a 2 man presence, and everyone gets what they need.
If you put a melee bottom, they don't get their XP that unlike ADC they DO need, and they may get less farm too. And you might not have scaling to fall back on.