First of all you could join in the Gambit, of course you can't simply join because you want to, since that doesn't make much sense but here in this gambit you have the very real stake of losing out on worshippers.
However you still can't send in the Hounds because you don't have the Cards its as simple as that.
But now you go: BUT WHYY? I have created the Damn Eagles why can't i use them?
Have you ever seen a war in which one side went: Ha i occupied the Crimea! and the other simply launched all its Nukes? No? Thats because its utterly mad and suicidal. As mentioned so many times before the concept of Escalation is in this game multiple times stronger than in other Games. Simply showing up as a God and nuking shit, will well, NUKE SHIT.
Showing up with all the Eagles with no other reason than YOLO i am Lu is a bit like USA nuking Russia for Crimea.
Imagine the entire Gambit system like the unspoken rules of engagement between the Gods. A system which ensures that all responses to any aggression or conflict are measured and proportionate.
To give you yet another better idea of why. Look at the Acterans. They basically appeared suddenly, then immediately began chocking to death on floating islands, then a huge hand came down from the heavens crushed the entire surrounding landscape and reformed it, then suddenly strange stones appeared out of the sky and a being which was so horrible and incomprehensible that they all are probably driven to gibbering madness appeared and told them to worship them for something they have no concept of whatsoever. Then an EVEN more horrible being shows up and a massive black spire with strange scrawls which burn themselves into their minds rises from the ground and he tells them the other is a liar, mind all speech so far is happening as mindspeak because the Acterans haven't even discovered language or anything really yet, so two horrible, maddening, seemingly omnipotent beings speak into their minds and start a fight. AND THEN Giant horrible beasts swoop down from the skies with eyes burning like the sun, sent by that horrible skyhand and occupy the weird stone thing which they were supposed to go into to thank one of the massive incomprehensible beings for being existent. Let that sink in, that is what you folks did on turn one to those people. They came into existance and basically will immediately go into gibbering maddened wrecks, mentally scarred to the point of unself-sufficiency, trying to hide away somewhere hoping that all of this will go away rather quickly.