Also resting your argument on an anime (which, while faithful, cuts lots of parts and/or changes events) instead of the actual source material (the visual novel) is kinda meh.
But yeah as stated above, the argument is invalid regardless, EMIYA would be obliterated by Ea if Gilgamesh had used it from the start.
The whole point of the fight is that Gilgamesh thought he would win using "just" Gate of Babylon, only to realize Shirou managed to counter it as an almost perfect 1 - 1, forcing him to take Shirou seriously and finally pulling Ea.
Which Shirou instantly recognizes as something he can't analyze, copy, or parry, hence why he sprints to cut Gilgamesh's arm before he can use it, otherwise he'd be doomed.
Gil greatest weakness is his arrogance, otherwise that fight would have been over in a nanosecond.
No form of EMIYA has connected with/seen the Root, or opened a new path to humanity (like star attribute servants). He isn't Alien or hasn't constructed his own NP (So no counterpart in Gate of Babylon),
So no it wouldn't be lore accurate to say he isn't weak to Enuma Elish, cause he is.