But I'm still puzzled why my current Safari browser won't work.
The reason is that bank needs to stop using the outdated and privacy-intrusive "user agent" query.
There is nothing wrong with your Mac's version of Safari, but your credit card company is using an outdated and ill-conceived method of determining browser requirements. Apple has essentially "frozen" the "user agent" query in a Quixotic effort to preserve what's left of user privacy these days.
The workaround for end users who are unable to update their Macs to a newer macOS version is to use a different browser, as you did.
If you are so motivated direct your credit card company's attention to the WWW publication User-Agent Client Hints (courtesy VikingOSX). It presumably describes the reason Apple did what they did.
In-depth background discussion here: chase wants me to update safari to 16, should I do that? - Apple Community