If The Void champs DIDNT Exist In The Game Then...

It actually woulda been pretty easy to tie gameplay directly into story while still removing the Institute of War.

You could set up the Void as a force that threatens to invade Rune Terra which already has all kinds of political stuff, espionage, etc going on in it. BUT!

To prepare the world forces to face the void invasion they put aside enough differences to set up a magical arena to train their best heroes and soldiers to face them (even while all nations are subtly calculating whose most likely to suffer the heaviest losses and who gets to take their stuff after they repel the void).

Also compare World of Warcraft with how the factions are at WAR but.....usually there's some "common enemy" that makes them begrudgingly willing to fight together (not like that outright stops them from fighting each other at all- they still have conflict in between fighting Demons, Lich King, Deathwing, or The Iron Horde, or those Great Old Ones (cough void/Lovecraftian Horrors cough that show up every now and then to bring madness and woe)

As a matter of fact, regardless of what you think of S2 whose putting out yes...another MOBA after HoN (Strife) this is almost exactly the kind of plot they went with but with kindgoms from different dimensions preparing heroes to face interdimensional invaders.