The condition is answering the summon they dont really need a wish, is just that if you dont have a wish for the grail you wouldnt want to be summoned normally. Being summoned is enough for rider of red wish, he just wants more opportunities for being a hero so it works (same way that Lancer wish in FSN was just fight strong people so being summoned is enough). Karna is more weird, summoning karna is really hard because he needs a bunch of mana to work so the fact that someone tries to summon him (and as he says someone with a weak body) is enough for karna to answer the call and the reason because only his original master is his real master for him.
Btw the grail isnt corrupted in this timeline, angra manyu was never summoned because in this timeline the einzberns summoned ruler (amakusa) not avenger, Darnic modded the system so they can summon a bigger rooster but nothing more.