I do not see balancing league as something hard

Champion balance is easy because of how isolated it is. You only have to deal with that single champion. Find what makes X too weak/too strong and change it to something better or more workable. (Worrying about how 'matchups' work is pointless. Nerfing Caitlyn might raise the winrate of who she counters, but Lucian vs whoever Caitlyn counters is still the same, so that winrate change is just a result.)

Item and rune balance is based on what champion classes you affect. This isn't really that hard, but it's a lot of more work than champion balance because you have to check every champion you affect. (Reducing Zeal Upgrades' item costs is a buff to crit ADCs, but you'll have to see if there's anyone who's already performing well and if this pushes them over the edge.) (Items include champions using X item.)

Map and Objective balance is the hardest as it affects everyone. The easiest way to deal with it is to divide it into the specific situations that deal with the Map and Objectives, rather than actually everyone. For example, reducing turret health would be a much big change to Ziggs than a split pusher. Changing an Epic Monster means you want teams to meet certain conditions to be met for fighting it, like Baron being high damage and tanky means you need to commit at least 4 people for an extended amount of time, which means you need significant map pressure.

When it comes to items, runes, monsters, and the map, unexpected things are bound to happen. Sometimes your community will point these out, and that should be listened to. Otherwise, they are unpredictable and they will happen, so just smartly deal with them accordingly.

The game also has a general balance: how bot lane, mid lane, top lane, and jungle all compare against each other. That's how you can tell the difference between everything being weak or a single thing being too strong in a role/class.