Having to level isn't supposed to add a sense of progression. It is designed to restrict what players have to learn so that they learn so they can learn everything the game has without being bombarded with tutorials.
At least in the early levels, it reduces the amount of information that a player needs to learn at a time. Not having all the Summoner Spells promotes learning because instead of having to choose two summoner spells from all the summoner spells, the player only has to choose two summoner spells from a smaller sample. Once the player gains a few more levels, the player will unlock new summoner spells, be interested in trying out these spells. They will naturally try all of the summoner spells without having to make deep choices on which summoner spells to take.
Same thing with runes. Lower rune count at lower levels means that the players can't take runes that aren't useful for players that don't know much about the game, and will have rune sets that benefit their less experienced minds. That assumes that they fall into the trap of buying the newer runes, of course.
Starting everyone at level 30 also has its own problems; since everyone starts at level 30, everyone would have roughly the same pool of people to play with. People who just started the game will likely be able to play with people who have been playing for months, if not years, and would be horribly outmatched against people who have been playing the game much longer. Having level progression ensures that they play against players who have similar experience to them (assuming they don't play against smurfs).
Moba's like League of Legends are well known to have steep learning curves, and League's way of restricting that learning curve helps the new player experience as much as Riot can help it.
