Gods summoning lesser beings for fun doesn't seem to fit League very well considering that there are near-godlike beings such as Azir and Xerath.
I think a better approach would be vying for a favour.
Riot has said that summoners are still part of the world, they just aren't all powerful or call all the shots. So, let us say summoners are a relatively common occupation (as common as a grocer, blacksmith, etc). Summoner's use magic to conjure weapons, potions, scrolls, etc, some even participate in wars.
Now let us also say, in order to be a summoner you need to be registered. Due to previous Great Wars from misuse of magic, magic-users are required to identify themselves/have a certificate/etc (Summoner Name tie-in). As summoners, you need to get better at your summoning craft. Problem is, there are no wars to fight and plenty of other people can provide weapons, potions, etc. To prevent summoners from being out of a job, the League of Legends is formed (a union/Summoner's Association).
This organization hosts contests, pitting summoners against each other in friendly competition. Summoners summon participants who want items or favours to participate in games that vary. A bunch of peasants that want new tools without paying for them approach the summoners and apply to participate in games. These peasants are summoned onto a soccer field where they face another group of peasants who want things deemed to be of equal value. The peasants of the winning team gets what they want, the losing team can try again for no cost.
So people can get what they want from summoners without paying anything if they win, so long as they have some physical skills to help during the games. Summoners get to practice summoning living beings and using spells during games (Summoner Spells tie-in). These games have been present since ages long past and thus are common all over Runeterra (this explains the different maps).
Sometimes summoners are presented with difficult tasks; brewing potions to restore fertility to a piece land; conjuring a magical weapon restoring an ancient artifact. For these requests, a death games with specific rules (gamemodes tie-in) are required. Summoners link minds with participants and simulate fierce battles on legendary battlegrounds.
Champions live in Runeterra and approach summoners (us) for requests which are extremely difficult.
Why would they approach summoners for requests? Maybe they are on the run 
Maybe no one else can understand them 
Maybe they are looking for something specific 
In any case, they make a request and are signed-up for a game. When enough roughly equal requests are collected, the champions are summoned onto a field. This explains why Garen and Katarina are on the same team despite their allegiances. They make requests from different locations but their minds are summoned onto the same field.
Occasionally, champions do not care who caters to their request (free week rotation). However, champions are picky most of the time and will not allow anyone they do not believe to be capable, to cater to them. Every time a death game is played, it adds to a summoner's reputation regardless of outcome. The influence earned in this way allows them to sway champions, convincing champions of their skills and allowing those champions to be summoned (IP + champion unlocks).
As we summoners participate in these death games, we gain more experience, learn to use specific runes for empowerment and gain mastery over our magical craft.
That's the best I can do right now.
