The way missions are currently set up encourages people who want to accomplish them to just play bot games.
I mean, I have time for 1, maybe 2, games per day. So if I want to beat a mission, I better win both of those games. Since league games are pretty close to 50-50, that's not going to happen consistently. You'd need 4 games per day in order to beat every mission if you're playing normal league games. That means around 3-4 hours allocated to league, per day.
The people who have that much time to spend on league each day will like missions, don't get me wrong. They also won't like the rewards from them very much though. They may get something like a legacy skin shard that they appreciate, but long term I doubt you're giving away a lot of paid content for free.
Missions would be a great way to help newer players (or people who don't play much) fill out their champion pools, etc... To get maximum value out of them though, I find myself playing beginner bot games to make sure I get the matchmade win on my first try (and the gold / kills / assists easily).
I think missions would be much better if they lasted for more than a day. Give me a week to win the mission and I'll keep playing the game as I enjoy it. Give me a day and I'm not playing a game anymore but really grinding bots just to get rewards. I used to do that when I first started (first win of the day each day on beginner's twisted treeline). I thought I was finally past that stage...