Eezo is both rare and common. Its current availability is determined by the I have two titles mom and nana and I rock them both shirt of the plot as far as canon is concerned. That said the amount of Eezo required to launch a small craft to FTL is much less than the amount of Eezo required for a dreadnought. This is quite clearly stated throughout the lore. So Dreadnoughts use more Eezo than a frigate or other small craft.
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As for the lore related to the Batarian Hegemony. We know they did it, so we’d murder I have two titles mom and nana and I rock them both shirt , even if we don’t have “proof” enough for the council. Archduke Ferdinand wasn’t killed by a member of the Serbian military, he was killed by a member of a Serbian terrorist group and look where that went. Humanity would be baying for blood especially when you add in the fate of those captured. We’d be deploying nuclear and biological weapons within hours, council be damned. Or we’d have revolutions, the Systems Alliance would be overthrown by its own soldiers at that point or at the very least fall into severe civil disorder if not war.