You are right, the new voice makes och sound like, I don't know what here. Something between e and i. For some odd reason, because the Voice usually gets och right, for instance the sentence Han och hon sounds just as it should (I can't link you there since there isn't a discussion forum for that sentence yet).
The slow audio is OK, although hon doesn't sound quite perfect.
In England people are taught to say "Him or her" "Is it him or her?" "Was it him or her?" In Sweden they would say -Han eller hon. Here are 190 million results from Google before jumping down my throat. https://www.google.com/search?q=%22Him+or+her%22&rlz=1C1PRFC_enSE784SE786&oq=%22Him+or+her%22&aqs=chrome..69i57j0l7.11089j0j15&sourceid=chrome&ie=UTF-8
As a mature, native, academic American I'm very sure that "him and her" (or "her and him", esp if the emphasis is on the female) is MUCH more common in spoken English than "he or she". And this is regardless of the context, even if strict grammar might require "he and she". This is similar to "Who is it?" "It's me." Almost no native American would answer with the pretentious sounding "It is I."
I can see a good pedagogical argument for using "she and he" in this lesson, but ruling out the most common usage would seem wrong.
I'm not against scratching the phrase entirely (and attaching a verb phrase to its replacement), but I am very sure, based on experience, that people will learn he/him and she/her wrong in both Swedish and English if we allow "her and him" as a translation. You have the benefit of being, as you say, a "mature, native, academic American", but please consider that literally hundreds of thousands of our learners do not.
It's not marked wrong, though. It's been accepted for five years. I just don't want to put it as a "best" translation because the system will then start using it to construct translation exercises.
I'll admit I should have written that much more clearly in my previous posts.
You meant to google "her and him", not "him or her", I assume. It's not a very good argument either way when "he or she" yields about as many hits, though.
It is "her and him" if the phrase requires a grammatical object for one reason or another. This phrase does not. In isolation, hon = "she", and han = "he", and that doesn't change with a conjunction between them. For instance, would you say "her and him went to the beach" or "she and he went to the beach"? You'd use the latter.
That interpretation requires specific context which is not present in the phrase. Each example you've given contains specific context. This phrase has no context - it's just three words. Allowing "her and him" would definitely teach people the wrong translations of those words.