There's a few reasons why I'm resistant to this idea:
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It would require a lot of extra effort from Riot to ensure quality control of such a system. That is, ensuring players aren't getting scammed, and being available to intervene in cases where players do get scammed, lied to, or cheated out of skins. Expecting Riot to uphold such a service is impractical, and If they didn't provide these services then dissatisfied users would collect and demand action or just quit the game all together.
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As it stands right now, there is not much incentive for people to hack league accounts. Hackers may be able to spend your current RP, but they can't transfer skins, icons, or champions off your account. Allow for trading and all of that changes. I guarantee there'd be a huge increase in accounts being hacked if a trading system was implemented, especially for streamers who often show off skins they have and whatnot.
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3rd party skin selling sites will become more ubiquitous. As it stands right now, these sites can only sell entire accounts, and so they don't get a lot of players interested. But if they could sell individual skins, they'd get way more customers (and how many of their skins would come from hacked accounts?). These sites are risky and unmonitored, so giving them a larger platform and access to the player base seems like a bad idea.
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Skins would have less value. Only one person of a group of friends would have to buy a skin for the entire group to trade it around and use it, or a group could all pool together money for skins and share them around constantly so it's as if they all own the skins.
I think Miles The Man makes some good restrictions on a trading system, but I don't think even that is enough to solve all the problems I've mentioned. In actually for a trading system to address all these issues, it would become overly complicated and ultimately ineffective.
As a player I enjoy the security of knowing that my skins, champions, and icons aren't going anywhere, even if I have to sacrifice the ability to trade with my friends as a result.