I'd say do away with runes entirely. Refund the IP values of all of them. Increase the base and per level stats of champions by a bit.
Rework the mastery system (again).
Remove a lot of the fluff and add more base stats, and do away with the "need x points to race down to the bottom of a tree, just pick whatever garbage is better than the rest".
Maybe give more points so you don't have to sacrifice too much utility to get the missing base stats entirely back.
I haven't put the effort into what this rework could become from these changes, as it's unlikely to happen anyway, but I'm sure these changes will be much better than having runes.
If they feel this takes away from 'people buying IP boosts', then good. People shouldn't need to spend money to get to the level playing field faster.
Though if they're that concerned, throw in some more chromas for IP.
As for changing the level up system.
Don't limit the points you can spend by level.
Introduce a new queue, called the "new player queue", which is a queue that will start after a certain amount of time has passed, no matter how many players have entered the queue. They can use the store, but cannot use runes or masteries, or any of the standard queues.
If 10 players are matched, it becomes a pvp game. If 1-5 players are matched, it becomes a vs AI game. If 6-9 players are matched, it becomes two separate vs AI games. This queue being separate will divide the new players from the somewhat more experienced players, which may not be many, so this max waiting time will keep them interested while they learn the basic mechanics of the game.
Then at about level 3 or 4, lock the new player queue, and give them a rundown on masteries, unlocking all of the points for it.
If we're still keeping runes, completely wipe T1/T2s, and give them some basic rune pages, which are empty rune pages that give you some typical T3 stats based on which one is selected, but cannot be modified. This gives the players the missing stats they would need, while still giving them the incentive to buy new runes to get the customization they want.
This unlocks blind pick and co-op vs AI, as well as all summoner spells, masteries, and rune slots.
Draft pick, ARAM, and custom games could be unlocked at later levels, as well as champion mastery, crafting, etc. Maybe restrict certain more intricate champions or masteries till they're a higher level. This gives the player nice milestones to work towards.
In order to make each and every level still feel rewarding, give some IP. Give a loot chest and/or a key at certain levels. They're already giving RP at... level 4 I think. So maybe some RP at certain levels too. IP boosts, some cheap skins, chromas, etc.
Both of these systems together would really help the game feel better for everyone involved, as well as make it easier for new players to get into.
Most other mobas don't restrict this much power over the course of a lengthy account grind.