I am 99.999% confident that Riot does NOT use banned accounts for personal use, and that 0.001% is only because it's not fully possible to predict what an individual person may do.
Without seeing more details (why you were banned, what error you get on log-in, what messages you get when you attempt account recovery, etc) we can't really say more, but there are a few possibilities:
- Someone else claimed that name when you changed it away.
- The account was marked as a stolen/sold account, and returned to another owner.
- Something else that we don't have enough information to say with confidence.
Jumping immediately to insisting that Riot is using banned accounts for personal play seems misguided at best, especially since every Rioter that I know would MUCH rather play on their own account and earn things there.
Send in a Support Ticket if you're curious, and someone may be able to help you.