Certain websites do not open on my 2011 Mac!

If you are having trouble connecting to websites with your older version of Safari, you probably just need to try it with a newer third party browser that is still being updated.

I suggest you take a look at Firefox browser or Brave browser. 

If you are using an ancient version of Safari on a 2011 Mac, then it is using the really old TLS security which would be incompatible with most current security conscious websites that have updated their security to exclude the deprecated Safari security standards.

That is why current versions of either browser recommended by D.I. Johnson will resolve those connectivity issues.

No reason to be upset with Apple. You remained on old hardware and use old software. Apple does not provide permanent operating system and browser updates for 12 year old Macs. That is your choice and would be remedied by current hardware, operating systems, and Safari browser updates.

Just to add, my better half still uses a 2011 iMac running High Sierra and websites, including banking sites, work fine with Firefox.