This is because it rankled that Mira got to be ramrod on that particular cattle drive while Elly herself ended up as pretty much Gabby Hayes. It bothers the crap out of any Patterson that someone else is the mad fool barking orders for the discreditable reason that other people tend to know what they're actually doing. They might piss and moan about family politics but underneath the huffiness and cattiness is the fear of their own flat-footed utter ineptitude. One can either live in denial like Elly or be like the rube she's married to and piss all over anything that doesn't interest him.
What this translates to is an eternal war against Mira because Elly won't admit that social rules that get in the way of her play-acting like a big shot are there to protect society from the crass and infantile. Just as her youngest bleats about being crushed by star power because her vanity keeps her from saying "Since I can't really love the person I see in the mirror, I have to piss all over someone harmless so I can go on not admitting it.", Elly doesn't want to admit that most of why she hates Mira is that she has the good socialist's need to jab her thumb in the eyes of the competent.