How about using a thermal fuse?

 

The fact that the Fan is turning does NOT indicate that
the Temperature of the Components being cooled by the Fan are in a safe range.
.
.
.

 

Because this is a fan that is blowing to the PCB and I want to eliminate usage of any other external components, somehow verify that with the electronics on the board level

If you don't want to use any external components, and just the circuitry on the mobo, the answer is _NO_ unless you've hacked something and can monitor motor current.

 

Waveform in the Figure is just an example of the current waveform I see.
How do I rectify the signal to get the average current?

Depends upon the amplitude of the AC portion of the waveform.
The signal could be coupled through a capacitor to block the DC portion, amplified with an op amp if necessary, and then rectified with a couple diodes.

 

Alternatively, a simple thermal sensor that is in the airflow could be checked at startup before the fan is spun up and you could look for a drop in temperature due to convective cooling. The problem with that would be if you also want to “watch” the fan and shutdown if it fails. I don’t think it would have a good response time to a failure but that’s just surmise.

If you measure the air temperature of air before it goes into the fan and again after the fan all you'll find is the same temperature. The fan moves ambient air. If it's blowing across something that is hotter than ambient then heat will be stripped away. So just measuring the temperature of the air without anything to compare it to would likely be a waste of time.

Using two temperature sensors and a comparator circuit you would be able to determine that there's a difference in temp at the sensors. One sensor would need to be just before the fan and the other mounted after the device that is in need of cooling. If both temperatures are the same then the fan is not spinning. But then comes a new problem: At start up before things warm up - air temp would be the same; and under that scenario the circuit would conclude no cooling is occurring and therefore be shut down.

 

Since a spinning fan makes noise - perhaps a microphone ? ? ?