"An High Priest of Good Things to Come" -- why "An"?

My impulse is to attribute it to two conventions: first, to a longer-standing custom of using a construct like "an historian" from British english — I encountered this constantly when reading Daniel Defoe's writings from..dunno, it was somewhere like the 18th century (somewhere 1500-1800..clearly not a historian).

Second, the inclusion of "an", particularly within a writing that isn't all that old (the Mormon version, which draws on clearly much-older and historically-evolving sources) and which is firmly rooted in American tradition, seems to me primarily to serve an oratorial purpose with a mild grandiosity — i.e., it just doesn't sound as ordinary as a grocery list or a periodical's articles.

So as to your broader question about the "an" modifying the larger message of the passage, I don't suspect it plays any other than a small reinforcing role in concert with the rest to collectively amplify the message that Jesus' personal growth in those matters far-surpassed mortal challenges — which, even so, were particularly fraught with travail, suffering, and death — by having to surmount existential challenges that could best the best among us, but then He also had to find His own spiritual path (on which the laity could never hope to follow) as He opened and navigated a nearly inconceivable bridge between mortality and divinity, to the betterment of all.

In short, I'd say it just serves to help impress upon all recipients of the message that this was anything but a normal undertaking which resulted in a theretofore-unparalleled moral ascension along His way to even more.

The wording of “an high priest” is archaic, and this creates a feeling of separation from ordinary language that you don't get as much with "a high priest."

Peter Shor pointed out in a comment that “an high priest” is a quote from the King James Bible, one of the most influential Christian texts in English. The King James Bible apparently uses “an” before many words that start with an “h” where we would use “a” in modern speech.

The use of“an” before words like “high” that start with a pronounced “h” followed by a stressed vowel should be distinguished from the use of “an” before words like “historical” which start with a pronounced “h” followed by an unstressed vowel. Some people still use “an historical” today in ordinary registers of speech, but I know of nobody who would use “an high” outside of religious or otherwise archaic language (setting aside people who drop the "h" in "high").