At this point if the HD is dead I would just leave it alone if you don't want to spend the money to replace it. Sadly your system needs the HD present as a sensor within it needs to be active of the thermal management to work correctly. You can see it here:iMac Intel 27" EMC 2309 and 2374 Hard Drive Replacement. Jump down to Step10.
If you are encountering problems that the system is slow to start up I suspect your real issue is the HD has become the boot drive not the SSD. Most likely you've tried altering the startup disk setting only to find it's not holding! So your real problem here is the PRAM battery needs replacing as you keep loosing the settings. You'll need to pull the logic board out to gain access as its on the backside I've marked the location here in green
Happy to hear the system is booting up with the SSD without any problems. Do to its age, I often find the battery has died which is why I assumed that was the reason you wanted to pull the dead drive as the system was taking forever to boot.
The system will boot, but the cooling systems fans will run wild as the sensor the system is expecting is missing! So the system goes into failsafe mode running the fans full out.
You're between a rock and a hard place.
While I fully understand pulling the drive as you don't want sensitive data to fall into the wrong hands when you sell your system. If you sell the system and tell the person they need a new drive as you pulled it, then I think you can hand them the system (make sure you write it on the bill of sale so there's no questions later).
If you or they use the system for long periods of time you do risk damaging the system without putting in a replacement drive or install a replacement sensor. Here's the part you'll need if you plan with or without adding a new drive to run the system: OWC In-line Digital Thermal Sensor for iMac 2009-2010 Hard Drive Upgrade. You don't need to install a drive to use the sensor.