Ultimately what I'm going to suggest comes down to philosophy, so if it doesn't align w/ your approach/needs I get it.

A theme of how you describe your interactions was to throw the support team under the bus, and/or highlight their failures to help you when interacting with an executive.

It's my experience that the executive has made intentional choices to deflect customer issues and that support team is their doing one way or another. Complaining about the support team to an executive will just put more pressure on the support team without granting them further support.

I'll get to a point in a minute :)

Another theme I see is one of using the system to your advantage. I have a fun similar airport story.

I knew my flight would get cancelled (weather) and I was about six hours early to the airport. I knew United had a partnership w/ Air Canada and the United staff wouldn't help me / the line was way too long because other flights already got cancelled. I went to the Air Canada desk and asked them to help since they had access to the same booking system. They told me to talk w/ United. While waiting in line again at United I called United on the phone and explained the situation and they immediately booked me an alternate flight on Air Canada that was direct to my destination, instead of through the one with bad weather. They also let me know that the Toronto airport I was in had a notorious rivalry between the staff of United and Air Canada where they constantly defer to the other team, supposedly because of spite.

So, my system.

Always take advantage of the system (non-human parts) - hit zero on the phone, try extensions if you dare.

My main approach when dealing with entry-level support people is to understand they have limited tools and their job is often to read a script that tries to get you to agree they helped. It's OK to deviate from that script, and if they can't help to ask to speak with a supervisor who can potentially offer a better option. When you move up the chain, it's important not to step on people on the way up. If they can't help you and you need to go around them by digging up the CEO via a cold call - I also think that's OK, but at that point you should be able to get what you want without throwing the failed support agent under the bus. If you want to give the executive advice, let them know that the support team seemed about as helpful as they could, but they didn't seem to have the training or support to offer you a fast resolution. Directly ask the executive to spend more time and effort investing in their support team and thinking of ways to offer improved support to avoid folks having to try to work around them. This frames it as a problem for the CEO to address with investment specifically, not one where they can just assume the support team did poorly and then go yell at a manager to improve morale with more beatings.

Anyways, always punch up is my rule. A second rule is to be deferential and try to take the blame for why you aren't getting what you want. This works exceptionally well at building empathy with support folks.

At one point in my life I missed about 50% of my flights, every time I'd be in line behind someone who complained about how slow security was, or the bag check, and they'd just get shut down. Every time I just asked if there was anything they could do to help me out given how absent minded I was for arriving late and not being prepared and I usually got the next flight out for free.

Anyways, punch up, use the system to your advantage, but don't direct your ire towards support teams when they fail. Also, if they've done a bad job it sends a better signal to fill out those NPS/CSAT surveys you get after the support interaction with a low score. This stuff will make it to execs, and comments that are left are often read thoroughly.