I sympathise with your "TL;DR" feeling (why oh why do academics - including the extremely knowledgable ones - so often make their otherwise-valid points soooo long-winded and self-referential? ...which is why I love the project started by Alan Alda - https://www.aldacenter.org/ by the way). In the author's partial defence though the initial "note to readers" text is follow-up to responses and updates, before the guts of the essay which follows (I think he could have more clearly formatted those parts differently in coloured boxes or such, so people could easily/quickly see where the "the original content" starts),

Firstly I say persevere with the essay if you can bear it (even if you need to skim initial verbiage) - there are a lot of profound insights, especially considering it was written 20 years ago as events were happening, and it ultimately answers your questions comprehensively. However for some quicker on-ramps about its primary tenet I was able to find from a quick DDG-search of "petrodollar currency war" that the rest of the reporting world is slowly catching-up (in many cases only now, 20 years later). Some top-links I found from that search (which I mainly just skimmed the beginnings of for context, so don't necessarily endorse entirely) are:

• https://markets.businessinsider.com/news/stocks/the-us-is-facing-a-major-challenge-as-petrodollar-loses-force-1032063614
• https://en.wikipedia.org/wiki/Petrodollar_recycling
• https://followthemoney.com/americas-petrodollar-system-a-timeline-of-the-rise-and-fall-of-the-us-dollar/
• https://www.washingtonpost.com/outlook/how-petrodollars-rearranged-the-world/2021/08/11/3474aee2-ca09-11eb-81b1-34796c7393af_story.html

The Wikipedia link about Petrodollar-recycling seems to have a nicely concise summary to answer your question:

How does it help or hurt the US if Iraq makes its the Euro or Dollar?

...and my very quickly typed (therefore far from accurate but hopefully high-level enough) answer would be something like this:

Following 1971 when the US forced termination of the Bretton Woods system (abandoned "gold-backed currency" for "fiat currency backed by smoke and mirrors"), by 1974 they became dangerously vulnerable due to over-spending on war (and some other endeavours) but found a quick-fix through petrodollar recycling ("buy loads of oil and the oil-producing country in-turn invests their profits heavily back into the US"). That was initially setup with Saudi Arabia but ended up being with all of OPEC, and because oil became the yardstick for international trade eventually the situation became such that the currency the world trades oil in became the de-facto "world trade" currency, and therefore the "international reserve currency". This creates a scenario in which "the US going under would take much of the world under with it" (generalising and summarising very crudely). That of course incentivised much of the world to protect the USD (and therefore protect the US from itself) in myriad ways and seemingly incentivised the consequent US administrations to hubristically spend wild/reckless amounts (especially on war) feeling like they are immune to "Consequences [tm]". The mantra was always "If you switch your reserve funds away from USD you will tank your country", but over time the expanding Euro-spending block of countries were becoming as big (eventually bigger) oil-buyers than the US, and Iraq switching their reserve to Euro turned out not only to be non-problematic but even "very successful". The US knew this would cause a chain-reaction of countries wanting to try the same switch to Euros (or at least be less phobic of considering it) so they needed it stomped out, while also finding other soundbite-friendly "reasons" for the stomping - screaming "look over here, look over here" so the mass-media would not notice the "petrodollar hegemony preservation" reason. WMDs was their gambit and it largely "worked" due to most people only listening to hot-button soundbites and retrofitting manufactured narratives to justify exceptionalism-fueled superficial knee-jerk responses. I think vanishingly few people would disagree with the fact that Hussein was a terrible, unforgivably criminal dictator, but not enough people asked "why are they suddenly only doing something about him now?".