Right, but as someone upthread mentioned, it’s jarring to the ear and the speaker to use a glottal stop at the beginning of a word. Much more tolerated in the middle of words.
In another thread today someone linked to an article about flour beetles. On said page there is this as a headline:
Is that a normal American use of “an”, because as a Brit that looks all sorts of weird to me.
Probably edited by a confused flour beetle
No, that’s just wrong. It’s probably an editing error of “How to detect an infestation of flour beetles.”
I thought as much. Thanks.
That’s exactly what I was thinking of. “Certainly, it can core a apple” 
No, it depends on whether that particular person pronounces the “h” in his own dialect. One of the significant differences between the American dialects (all of them, AFAIK) and the Canadian is whether the “h” in “herb” is pronounced. So in Canada, it would be, “a herb”, while in the US, it is “an herb”.
The “a eulogy” is a nice example I had not noticed. Similarly there is a mathematical concept of “a Euclidean domain”. Another “u” example is “a university”. And “a gram is a unit of mass”.
And, if I recall correctly from an A Way With Words podcast, British English speakers pronounce the “h” in “herbal” because when they adopted “herb” from French, they pronounced the “h”, which the French didn’t. American English, rather than adopting “herb” from British English adopted it independently from French and kept the French pronunciation. Or something like that.
This whole a/an thing drives me nuts because you hear American media saying things like “an historical” and “an hysterical” all over the place these days. I’m surprised they don’t say “an horse” yet.
Huh, I didn’t know that about Canadian dialect.
However, your latter examples are all the sorts of examples in line with the OP. You use an “a” because next word doesn’t start with a vowel sound. They start with a “y” sound.
Sort of makes me wonder if many native New York City folks would say, “a human” or “an human”. 
I’m not sure what you mean. Whether you drop the h (as a New York accent sometimes does) or not, human would still take “a” (yuman vs hyuman).
Damn! I didn’t think that one through. Did I?