Can run my Intel iMac without the fan

In general it is not a good idea. It appears that you forgot to replace the thermocouple with the hard drive. That would be the correct solution.

But, there are apps like iStats that will report the temperature of the CPU to you. run one of those and see if your CPU runs excessively hot during operation.

Ah, yes, the classic "if it's noisy, let's just unplug it" solution. I get it, fan noise can be annoying. But, you see, the fan is there for a reason. It's like your personal butler for your iMac's CPU, keeping it cool while it keeps running your email, music, and your adventures through the digital cosmos. In your case, you've just fired the butler.

So, what could go wrong, right? Well, let's start with the obvious: heat buildup. Even if you're not running any demanding applications, components will still generate heat. Over time, that could degrade your hardware faster than a teenager devours a pizza.

Your iMac's design incorporates the fan for a reason—much like how your car has brakes. So, if you don't see the point of them until you actually need them, it's often too late.

You might get lucky. Maybe you won't fry anything. But if you do run into issues, let's just say you'll have a hot, non-functioning paperweight on your desk. Given the year of your iMac, it's already considered vintage, or should I say, a digital dinosaur. You really wanna risk sending it to an early tech afterlife?

If the fan noise was unbearable, there are other ways to solve the issue without ditching the fan altogether. For example, you could look into software solutions that help control fan speed or even consider replacing the old fan with a quieter one.

Trust me, that fan isn't just there for ambient white noise. It's like the air conditioning in a car; you can live without it, but why would you want to? Especially when it could cost you a lot more down the road. So, proceed at your own risk, my friend. You've been warned!

I can't warn you any better than Jps_13 did.

Since you did the installation your self check with OWC (MacSales.com) on what you need to replace temperature sensor to the drive (or the one you unplugged). I imaging it will also need some thermal paste to bond it to the drive.

Another warning: if you use Safari and have a lot of windows open that will put quite a load on the GPU an CPU causing the fans to ramp up for a while. Without the fans you're looking a premature failure. And with a 12 year old iMac that will be much sooner than later.

If you need to do the same at a later date get an external SSD, clone your boot drive to it and boot and run from the external. You can take the external with you much easier when you move on up.

You are rolling dice by disconnecting the fan. There was an issue on older iMacs when replacing the hard drive with an SSD. There was a temperature sensor cable for the hard drive and when you removed the hard drive, most likely the temperature sensor was left disconnected. That would have resulted in the fan turning on. The solution back then was to place a jumper on the connector. I suggest you take a look at that. In addition, as you do have a 2012 iMac, do a thorough cleaning of the interior of your iMac as it must have substantial dust and other contamination on the cooling fins and the system fan since that will also contribute to an elevation in system heat.

It could also be that a different temp. sensor is disconnected or damaged and causing the uncontrolled fan speed.

Does HDD Fan Control show the read outs for all of the other iMac's temp. sensors?

If it does, then I would look for a wonky reading from all of the different sensors.

If it does not, then I would try > https://crystalidea.com/macs-fan-control

You can drive your car with no coolant in the radiator too, but sooner or later something's going to meltdown.

That should be enough. Therefore try the following:

1 - Reset NVRAM or PRAM on your Mac - Apple Support (for Intel Macs) and Mac startup key combinations - Apple Support (for both types)

2 - How to reset the SMC of your Mac - Apple Support

See if the above will kill the always on fan.

I am running High Sierra OS 10.13.6, the latest OS for this iMac.

A 2012 iMac wil run up to macOS 10.15 "Catalina." See:

macOS Catalina is compatible with these computers - Apple Support

With the move to the thin-case design in 2012, iMac cooling was compromised. 2011 models had three fans; yours has but one, and needs it. Could it be the one you installed is not working?

Note the thermal sensor included with this install kit:

https://eshop.macsales.com/item/OWC/K27IM12HP1TB/