Regardless, I accepted the bank's advice and checked things out. First, as I had done with the Flash video problem, I shifted from Safari to Firefox. The page loaded just fine in the alternative browser. This was encouraging. At least I knew the page still existed. Now I just needed to convince Safari of this fact.
Returning to Safari, I went to Safari -> Preferences -> Privacy. I confirmed that the "Block cookies" option was set to "From third parties and advertisers." This was the same setting that had been in place for months, even when the page was loading correctly. So I was not optimistic about cookies being the cause of the failure. Still, as a test, I changed the setting to Never. Surprise! The Bill Pay page now loaded in Safari. Even stranger, when I reverted back to "From third parties and advertisers," the page continued to load correctly--and has done so ever since. The cookies setting was the cause of the 404 error after all!
As far as I can determine, Safari was behaving as if "Block cookies" was set to Always, even though that was not the setting indicated in Preferences. Temporarily changing the setting apparently got the glitch to correct itself. Indeed, I have since read of other instances of "spontaneous" shifts in cookies settings in Safari--although usually without the mismatch between the actual and displayed setting.
My general advice here is to check Safari's cookies settings whenever a webpage unexpectedly refuses to load. To test whether cookies are behind the error, set "Block cookies" to Never--at least temporarily. If simply enabling cookies does not resolve the issue, you may need to delete the cookies for the entire site. To do so, click the Details button from the Privacy screen, search for the site's cookies item(s) and click to Remove.
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