I don't know about that since so far Gilgamesh doesn't really seem to care about the Grail beyond that all treasures belong to him and judging it's worth. Lancer doesn't seem to want it for himself just to hand it over to his master since when Rider gave his proposition he didn't say he would obtain it rather his master, yes you could say it means the same as him wanting it for himself but his words doesn't indicate that he wants it. Caster and Berserker are crazy though they might want it kinda hard to say at this point and Assassin doesn't seem to care about the Grail either otherwise they wouldn't agree to what Kirei is doing.
We also have Assassin from the 5th war that didn't care about the Grail only about facing worthy opponents. Archer didn't really need the Grail either so you could say he doesn't care about it in the first place since it seems he never intended to win. Lancer gave up his chance to obtain it, if he really wanted it he wouldn't just throw it away and no idea when it comes to Rider and Berserker. So from what I've seen it seems a lot of them don't really care for the Grail.
You could be right about them wanting the Grail but how do the masters know which hero actually wants the Grail, since they go through all the trouble of finding relics etc to summon them it would also require them knowing they have a need for the Grail and I don't know if stories from the past is enough to judge if someone wants the Grail. If it was the Grail that picked the heroes I'd understand but not when it's the masters that pick.
It seems kinda weird to say they only exist in the Grail war at the very least Archer from the 5th war existed beyond that and after his death as well judging by the movie so if he's an heroic spirit like the rest there shouldn't be any reason why the other heroes can't do the same. And then there's the deal with Gilgamesh.