After numerous experimentation with multiple devices and multiple operating systems, I have reason to believe the problem is actually twofold.
It's not a scam. It is the result of particularly ill-conceived page code, and it's something its developer needs to fix.
I see the homepage load for a brief second, then POOF, this page appears.
I was able to duplicate that, but only a couple of times.
As it turns out that particular website uses a Javascript to (among other things) measure the response time for loading a page element for the sole apparent purpose of deciding whether it will load or not. If it takes too long to execute that Javascript, it displays that error page.
The substance of it is incorrect in that its allegation of an "unsupported browser" lacks justification. There is nothing wrong with the browsers you have been using, other than the fact Safari is taking longer to execute that Javascript than the page code expects.
You were right after all. It's your iPhone, but it's also webpage code that's just a bad idea. You may be able to get the page to load using a browser that executes that particular Javascript fast enough to placate that ill-conceived "browser check" that serves no apparent useful purpose.
What's worse is that the link to update Safari redirects to a site notorious for hosting malware in the past. It could have just as easily redirect to the correct Apple website. Not to worry — no malware can affect your iPhone — but it illustrates how poorly thought out that particular page code happens to be.
All of this suggests that perhaps your iPhone isn't working as fast as it should, but that's a subject for a different question.
For a workaround try another browser such as Brave:
https://apps.apple.com/us/app/brave-private-web-browser-vpn/id1052879175