You keep bashing software that operates and monitors your fan control and even say that you have to listen to the fans roar full speed on boot up. That is not true at all.
Like hell it isn't. The fans roar till the extension loads. Remove the temp sensor from a 2009–2012 iMac and you'll see I'm right.
If you're a gamer who uses it in addition to the onboard temp sensor, that's different. This, however, is a 2011.
Can it boot successfully to internet recovery (not sure if a 2011 iMac can DO internet recovery)?
You already know the answer to that. The 2011 iMac can't do internet recovery. You can option-boot and reinstall the current OS and/or restore from a Time Machine backup.
Likewise, one can go into the Repair partition and run Disk Utility. Try to repair the volume. Keep running till it is green but if you get the same error in red three times, give it up. If the error in red keeps changing, that's good and you'll probably get to green eventually. DU often fixes problems one at a time — I've run it over 50 yimes on a drive to get it into the green... Except it doesn't really repair the basic problem; it writes around it. You hope it lasts long enough enough to get the data off. I've had that last anywhere from 15 minutes to 2 weeks.
Error messages on an old HDD are a 100% indicator of mechanical failure. I've worked on and replaced hundreds of these. Not one passed a complete SMART test even if they exhibited no symptoms to the student. OTOH, I have a small stack of HDDs from 2007–early 2009 iMacs that are still good. I use them for testing this and that.
A 2011 iMac should boot from a USB startup drive with any OS from 10.6.4 – 10.13.6, 8GB or larger. Likewise, a USB or Thunderbolt external. It may or may not do so from a Firewire drive — if it shows up in an Option Boot, there's a chance.