Why don't we utilize back EMF power?

The energy stored in a relay coil is so minimal it is not worth the cost of the extra componentry to reclaim it. We just accept that switching the relay occasionally uses a small amount of energy. A typical relay might have an inductance of 50mH a DC resistance of 400 ohm, so the energy stored in the coil is about 20uJ, if it operates once a second (far more than it would in a typical application!)

the actual power loss due to not reclaiming the stored energy is then 20uW. Far more is lost as heat because of the resistance of the coil!

The voltage across an inductor doesn't represent energy, but the rate of change of energy stored in the inductor. The voltage is high because the small amount of stored energy is very quickly reduced to zero. The energy stored in an inductor actually depends on the current.

The back emf is actually used all the time. Any passive linear filter that uses an inductor relies on energy storage and release from an inductor. All the common switching regulator style circuits rely on energy stored and released from an inductor, plus countless other applications.

That sort of thing is done in the highest efficiency switch mode power supplies, which sort of turn on and off tens or hundreds of thousands of times per second. Current from a snubber is shunted back into the primary of the supply. The amount of energy recovered is miniscule, but in a market where 96.1% efficiency will get the sale over a 96.0% product, the extra complexity is worth it.

For a consumer device, I don't know what would be done with the tiny amount of energy captured after turning a device off. It would be hard to store for next time, as capacitors self-discharge. And feeding the power back into the wall outlet would involve complexity, cost, and safety issues, assuming the circuitry didn't consume more energy than the spike put out (quite likely).

That sounds like a nice little demonstration you've built.

We do use back-EMF, a little bit. Most of the time, though, the amount of power available isn't enough to be useful.

There are exceptions. Electric light rail mass transit systems (subways) use their motors as generators for braking. That's back-EMF being pumped back into the third rail.

Some hybrid automobiles will do the same thing, charging the batteries when the driver backs off of the "gas" pedal.

It's called "regenerative braking". But it's really a back-EMF application.

That's how fluorescent tubes work. They charge a coil (choke) through the starter, then when the starter heats up & goes open circuit, the back emf jumps across the tube to ionize the gas inside & light it up. It's also how a car's ignition system works.

Remember, you won't get more power out than you put in, you're just changing the voltage by using a different resistance for the discharge path.