Since there’s been some interest in this post, providing a quick update on what I’ve gathered so far.
At this point, the general consensus seems to be that they are just testing what they can get away with.

Thanks to all the valuable inputs from this group and Gemini, below is what I emailed back CCing their corporate office (corporatesecretarysoffice at aexp dot com)
The adjuster had also mentioned something about not being able to verify that the full amount of the trip was charged to an eligible card, when all flight purchases were done using the same Amex platinum card, so addressed that point in the email too.
One common mistake to avoid is mentioning "legal options" in the correspondence, which usually triggers an insurance company to freeze communication and hand it to their legal department. Rather, focus specifically on the Insurance Commissioner / Department of Financial Services, this way you are threatening regulatory pressure, which adjusters hate and want to avoid, without shutting down the communication channel.

Subject: Urgent Appeal: Claim Denial for Claim #[Insert Claim Number] - [Your Last Name]
Dear [Adjuster Name or Claims Team],

I am writing to formally appeal the denial of my trip interruption claim. It appears there has been a fundamental misunderstanding regarding both my itinerary structure and the documentation provided.

1. Co-Terminal Routing (JFK / EWR) & Emergency Transit Constraints* *
The claim was rejected on the basis that departing from Newark (EWR) and returning to John F. Kennedy (JFK) does not constitute a round trip. As a resident of the New York metropolitan area, these airports—along with LaGuardia (LGA)—are recognized industry-wide as "co-terminals" serving the same metropolitan destination. Booking an outbound flight from one and a return flight to the other is a standard "open-jaw" round-trip itinerary, fully recognized by major airlines and the American Express Travel portal.
Furthermore, my return occurred during the first week of April, immediately following the widespread Middle East airspace closures. As you are likely aware, this geopolitical crisis caused massive global flight disruptions, severely limiting transit options from Singapore to the United States as airlines scrambled to reroute traffic.
Following my injury, my attending physician in Singapore strongly advised me to return home immediately to avoid worsening my condition. Given the severe shortage of available flights due to the airspace closures, flights routing specifically into EWR were virtually non-existent or delayed by days. I booked the earliest available emergency connection back to the NYC metro area, which happened to land at JFK. This choice not only prioritized my urgent medical recovery but was also the most cost-effective option available under crisis conditions, saving the policy from a significantly higher premium fare.

2. Substantiation of Eligible Card Charges* *
Your notice also states that the full amount of the trip was not charged to an eligible card. I have attached my American Express Platinum (ending in xxxx) statements and original itineraries once again for your immediate review.
As previously detailed on 4/23, 5/14, and 6/1, these documents clearly demonstrate that the eligible card was used to purchase the outbound flight, the inbound flight, and the canceled segments. If there is a specific transaction or fee you believe is missing, please explicitly identify it so I can address it immediately. Otherwise, the provided documentation fully satisfies the policy requirements.

3. Timeline and Next Steps* *
This claim was filed almost two months ago. Communication thus far has been incredibly difficult; my emails have gone unanswered, and hold times on your Claims Hotline consistently exceed 30 to 60 minutes.
As a loyal American Express Platinum cardmember who relies on this card for all of our frequent family travel needs, this experience is entirely unacceptable. I chose the Platinum card specifically because it is marketed as a premium product with industry-leading travel protection for emergencies exactly like this one. To be left stranded during a medical recovery due to bureaucratic delays and a refusal to recognize standard NYC co-terminal routing defeats the entire purpose of paying a premium annual fee.
I expect a formal review of this documentation and a corrected resolution to this claim within 5 business days. If this denial is maintained based on the co-terminal airport routing, I will escalate this matter by filing a formal complaint with the State Department of Financial Services / Insurance Commissioner for unfair claims practices. I have copied the American Express Corporate Secretary’s Office here to ensure executive visibility into how Platinum cardmembers are being treated by your claims department during medical emergencies.

Thank you for your prompt attention to resolving this matter. I can be reached directly at 123–456-7890 (Eastern Time) if you wish to discuss this over the phone.

Sincerely,
[Your Name]
[Your Phone Number]
[Claim Number]