Honestly I don't think a required number of wins would help much. I got to level 25 or so in the old system, playing mostly Teemo, before anyone bothered to explain what Ability Power was. Last hitting, intentionally? No one thought to drive that home until even later.
The problem lies with what and how new players are taught about the game. Too much of the tutorial is spent covering things that anyone who's played a video game in the last 15 years probably knows or can figure out in 20 seconds, making it feel like a waste of time. After going through the dull tutorial and getting your
to the enemy nexus, you're quite done with tutorials. Any other information new players should know is ignored because of how bad the first tutorial is.
Fixing the tutorial, possibly adding a reminder of some of the basics on the side bar if someone's not CSing enough or something, will do more to help new players to Ranked than raising the champion limit or adding a win limit.