The durable fix is to stop relying on their page: run your own probe against the endpoints you depend on, from outside your application, on a schedule, and keep the history.
This costs very little and it changes every future conversation. Instead of "we think you are having problems", you arrive with a graph of your own success rate and latency against their API going back months, with a visible step change at 09:14. That is not arguable, and it is also proof it is not a change on your side, because the probe is not part of your application.
Two details that make it worth having:
Probe from more than one place if you can. Two regions turns "it is broken" into "it is broken from here and fine from there", which is exactly the sentence that gets a partial outage taken seriously.
Record the request id on every probe, successful or not. Then when something goes wrong you already have the identifiers rather than scrambling to collect them while it is happening.
And the general habit: subscribe to their status page updates, but treat the page as their opinion, not as data. Your own measurement is the only thing that tells you what your users are experiencing, and it is the only thing that ever gets a vendor to look properly.