To Wheel64:
This is an update to my original post and referencing your problems. Like you I suspected it was not just Safari 15. So.....
I reinstalled my Hard Drive -using the Restore feature - from my Time Machine Back-up of a few days before, i.e. before downloading the Security Patch and Safari 15.0 at the same time on Sept 21. which then caused all these problems.
The re-install took about 3 hrs. but it sorted everything out, and no issues. NOW the good part...
I decided to move step by step to update, to be current.
So I only allowed Apple to download and install the Security Patch, on its own. Once installed I checked everything was OK. And everything worked OK as before, but I am still using Safari 14.1.2
So, then and only then I downloaded and installed, ( as separate instruction) Safari 15.0 and nothing else. This item just took less than 5 mins. and my system didn't need to re-boot.
Again I tested everything and everything worked... no issues.... all websites opening their log-in pages.
My only thought is that installing the Security Patch and Safari 15.0 simultaneously caused the bug on my MacBook pro to prevent accessing -in my case - a whole host of financial institution log-in pages. However by downloading these 2 items separately allowed my system to update with no errors or issues.