Actually, it's a fantastic idea, one that Counter-Strike: Global Offensive just went into beta for. I have already seen an increase in both positive behavior and competitive like-mindedness.
I'm hugely fond of people having to tie their accounts to more concrete and daily essential objects apart from emails. Not only does it permanently get rid of cheaters, but players who consistently make new accounts and continue to be negative.
EDIT: If you cannot afford a prepaid phone and one text message, you probably cannot afford your internet connection, either Thanks for pointing out that I wasn't being inclusive of college students, LostFrost! For those people who are in college and don't have bucketloads of money, Ebay has $20 prepaid phones with no contracts/requirements. Additionally, Skype numbers and other numbers should be blocked from being able to subscribe to this service. If you're under the age of 18 and do not have a phone, you can ask your parents to connect theirs. Here's where it gets really awesome. Your parents confirm and link your account to their phone, and when you get a permanent ban or other similar punishment, guess who the text message goes to with the chat log?
EDIT: It doesn't have to be a mandatory service, either. It could be opt-in. See my response below for that, as it's closer to the model that Valve/Hidden Path are taking with CS:GO.
I applaud Hidden Path for their recent move to this system, and their past decision of having Overwatch, a spectator program specifically designed to detect cheaters and/or griefers. There are cases where, similarly to a Battle.net authenticator, people might need to transfer ownership from one number to another. That's absolutely fine, but if that account gets banned on that number, it cannot be used again in the future. The only issue is that you'll have to provide your name and current service, that way if you get a new phone number, other people who inherit your old number aren't barred from creating an account or additional accounts.
Another caveat to this system is that for high-elo players, limiting the amount of accounts a user can have will significantly impact the chances that they get a game in a decent time. However, I believe that this is a good reason for Riot to consider improving upon the long queue times as they have been, now just with a much higher priority.
If your account gets banned for any reason and you lose the appeal in a ticket, you should not be able to come back and play this game. So your options are to either, 1. appeal your ban and win, or 2. Get a brand new phone number, which will cost you quite a bit of money. When you start attaching every-day services to accounts like that, people's attitudes change big time. If it's completely free, you have nothing to lose, and I don't like that approach.
There are great ways to approach this system: You can apply it to every game mode, or you can simply apply it to ranked-only matches.