You're never going to play 123 champions all the time
Speak for yourself. As a veteran of both hon and dota, I'm a frequent player of "all random" mode. In both games, this means you get a different experience every game. You may say that you'll never master all of them by doing it this way but that's besides the point. By playing a character you learn all of their abilities and how they work, what their cool downs, mana costs, attack and ability ranges are etc. Doing this at least once for every character gives you a much better grasp of what you're up against than trying to learn only by way of opposition.
The 'all the time' part of my statement was quite an important bit.
Sure, ARAM exists - but like I said there's an always-changing array of free champions, so it's not like you'll never get them. And again - if you're just starting out, there's going to be some time before you even get around to trying every champion for the first time. Even if you go with ~30 minutes per game(below the average length in normal modes, slightly above average on ARAM) - you're looking at upwards of 60 hours of playtime just to try everyone once.
You can complain about F2P games all you like, but I don't think for a second we'd have had as many new champions released if they'd been on a pay-once model. I don't think that's exploitative, just different.
-edit- before my play patterns are criticised again, I should say I've been playing LoL to varying degrees of intensity for about 5 years. I think it's more accessible in terms of champion acquisition now than it was then.
It dampens the competitive scene not to have access to all the characters. It's difficult to join a team and play support if you don't have access to the support character you need to play. And since the most recent Dota world champion was 16 years old at the time, kids matter. And kids can't purchase the characters they need to purchase, since they have no money.
There's a thriving competitive scene in both games, so maybe the effect isn't so bad in practice. But if you want to play soccer, you'd find it difficult if you didn't have access to certain parts of the field unless you pay money. That seems like an accurate description of League's model.
The central issue is this: If you want to follow the strategy of making a new game competitive and then funding a yearly tournament, your game is more likely to fail if it uses League-style "pay for access to the competitive landscape."
League works for League. It's largely thanks to history and timing that League grew to such a degree: When League launched, there was no Dota 2 and no HoN to compete with them. League's model is less likely to be replicable to a new game.
All Random in hon and dota is played on the standard map with standard rules. Each player gets a random hero chosen from the entire pool. The only restriction is no duplicates. This mode can be played from the very first game on a brand new account.
ARAM in League is totally different. It's not even available to players until they have purchased/unlocked enough champions to have a pool, a situation that could take a very long time to resolve for players who have no money to buy them.
You gave an estimate of 60 hours for a new player to try all 120 heroes in hon or dota. That could be accomplished in one week by a dedicated new player. How many hours do you think it'd take to unlock every champion in league? Thousands upon thousands, at least.
Edit: I'd also like to point out that in hon/dota you get to play a hero at their full power and effectiveness from the very beginning. This means that a veteran of one game who switches to another is not at a disadvantage, unlike a new player to league playing against people who have played long enough to max out their summoner level, runes, and masteries.
Speak for yourself. As a veteran of both hon and dota, I'm a frequent player of "all random" mode. In both games, this means you get a different experience every game. You may say that you'll never master all of them by doing it this way but that's besides the point. By playing a character you learn all of their abilities and how they work, what their cool downs, mana costs, attack and ability ranges are etc. Doing this at least once for every character gives you a much better grasp of what you're up against than trying to learn only by way of opposition.