https://support-leagueoflegends.riotgames.com/hc/en-us/articles/360037264394-Riot-Accounts-Changing-Your-Username-FAQ
"WHY ISN'T IT FIRST-COME, FIRST-SERVE? SHOULDN'T THE OLDEST ACCOUNT GET THE NAME?
We looked into these systems (and more) and found that they presented their own set of issues. For instance, if three players have the same username, and we flag that the “first” or “oldest” gets to keep it, that still doesn’t solve the fact that some other players out there also have that name. When they try to log in, the accounts system can’t know if it’s you, or if it’s a different person trying to log in, since you all have the same username, which means we’re right back to square one. We can’t go around just changing players’ usernames without them knowing how to solve this, because that risks completely locking players out of their accounts for good. In the end, first-come, first-serve created several problems that risked players losing their accounts, and we didn't want that for anyone."
Im not sure I understand your issue. You are refusing to change your username, hence are not playing anymore? You are indeed free to refuse but that means you cant play anymore.