(I apologize in advance for the word wall.)
Okay, I highly agree that they should make all Summoner spells available at all levels. With that said, I don't think they should do the same with Masteries and Runes. Summoner spells are extremely helpful in-game and they can literally change the tide of the game with Smite on JG'ers. Just last night I played a game with my level 6 friend to help her level up and I, like always, called top. We had a Sona mid, a MF and a Tahm Kench bot, so she was stuck going JG with Pantheon. We ended up winning the game (mostly because of the Sona), but she had an extremely hard time leveling. I believe we won the game with everybody level 18, Kench at 17, and she was at level 12 still. I just feel like that's ridiculous. She can't JG because she doesn't have Smite because what, she hasn't played long enough? Then again, though, I do still feel that, if for nothing else than for marketing purposes, Riot needs to keep some leveling-based perks in the game that still affect the game itself, like Masteries and Runes.
Masteries are important to in-game, sure, but nowhere near as much as Summoner spells, so I feel like that's something they could leave as a leveling-based perk. Runes, then, are even less important to in-game success. Again, they are important and they're extremely helpful, but there are people who win games without them. And when I say in-game success, I think I mean more that, without, say, Smite, there's almost no way to correctly work your position in the game, whereas Masteries and Runes aren't going to hinder your ability to JG. They might hinder your ability to JG WELL, but you can still JG correctly without them. They aren't a necessity for fitting your position on the map. Now I fail at League and know next to nothing about its mechanics and all that jazz. However, I'm pretty sure I'm not wrong on this one. If I am, I apologize. However, I still say Summoner spells are probably the only thing that would hinder someone's ability to simply fit their position on the map.
I would also like to address the fairness you mentioned. You say that the Masteries and Runes give an unfair advantage to some because they've played longer, but that's not the case. Masteries and Runes are a player-chosen thing. Yes, the slots to get them are leveling-based and the quality of the Runes are leveling-based, but the player still chooses which Runes and Masteries to use both in their sets and which set they use in each game. Now this already kind of disproves the unfairness only because just because people have the slots, doesn't mean they're going to use them correctly. Veigar hurts a lot with his AP, so if you build MR, you can defend against him a little better. But, if you're doing blind pick, you don't know you're up against a Veigar, so say you build AP as well. Now you're just making it harder for yourself. The more AP you have, the more damage he does to you with his own AP, so whether or not your item build consists of MR, your Runes and Masteries still give AP and that's going to put you at a slight disadvantage. Basically, just because somebody has the potential and capability to be at an advantage, doesn't mean they know how to or will do so.
Now I addressed that, but I should also say the other part of it for safe measure. I main Gnar, who is typically built either AD or Juggernaut or both. Of the two games I built AD, I lost one and won the other. Of the 15 games I built Juggernaut, I only won 3 and lost 12. Now, of the 18 games I played building AP Gnar, I lost 10 and won 8. Also, since I fail at this game, people who main Gnar and are good at League would probably do a lot better when doing the same build as me. Where I'm going with this is that most people build a certain way against Gnar because he generally goes AD or Juggernaut. People on my team also get mad at me for building AP because something or other about scaling. So people don't expect me to build AP against them. Therefore, when I do, it throws them off and they, for a second, can't adapt fast enough, giving me enough time to get in a kill or at least base them. So whether or not they have more or less Masteries and/or Runes than me, I still built mine different than they were expecting, so they weren't able to fight me as well. Putting this in a more objective setting, if someone were to build something the enemy wouldn't expect or that isn't all that common, even if it only works a little bit, it will still throw the enemy off enough to give you an advantage, so the amount of Masteries and Runes someone has doesn't matter near as much as Summoner spells.