It would have said "each friendly Hoth objective" if that was the intent. In this template, "a" doesn't mean "each". I think we're all on the same page though.
It's not as unambigous as you make it out to be.
In the English language, "a" is an indefinte article. If they wanted "a" to mean that "the first, and only the first, friendly Hoth objective" attacked triggers the ability, which seems to be what you're suggesting, then they should have used the definite article "the" to describe the objective. Likewise, they should have used the adjective "first" to signify which objective it refers to, similar to how I phrase it above. But they don't do that do they?
The limiting adjective "first" is only designating which character gets destroyed, not on which objective the ability triggers. And that is the problem with the current wording.
The fact that they use a definite article and adjective to describe which unit is destroyed (notice how they used the definite article "the" and the adjective "first" to signify which character is destroyed: "the first enemy unit"), implies that by design, they purposely did not want to specify and also did not limit which objective triggers the ability, other than it having to be a Hoth objective.
The way it is written, "a" signifies any Hoth objective. So as Dbmeboy cleverly points out, If Vader attacks your "Echo Base Defense" objective and is the first character attacking that particular Hoth objective this turn, Vader get destroyed. I think we can all agree on that. But, if later in the turn, Emperor Palpatine then attacks your "Hoth Operations" objective, and is the first enemy character attacking that friendly Hoth Objective, why wouldn't he be destroyed too? He meets all the conditions of the enhancement as written. He's the first enemy unit attacking a particular friendly Hoth objective on that turn. I agree with you that had they used the phrase "each friendly Hoth objective" it would be clear and unambigous too. But they didn't specify one way or the other, and that is the problem that is creating confusion for some.