When I first posted in this thread,idea/experiment had not been done and it was still in everyone head. Now some stuff has been done to enhance our use of the NFC Ring.

Initially, I thought of a modificaton that would enable the nfc sensors even when the screen is off. I don't know how much battery drain that would cause, but maybe a lot because the sensors always need to send pulse ?
Of course, better methods are available : when the Moto X was released, it pleased everyone with it's active notification system. With the good ideas, some users created programs to duplicate Moto X features like "Gravity Screen", "Active Display", "Ac Display" and so on.

The software "Gravity Screen" allow to turn on the screen when it detect that you take your phone out of your pocket. It does this by using the proximity sensors (used to turn off the screen when you are talking to someone to minimize touch event) and this method is very good for the battery. Then, as the screen is on, the nfc should start working and your phone will unlock itself in your hand. Nice! I take my phone and it's already usable.

There is other software than this one to do the job, and it can also be done using tasker, but with tasker the proximity sensors detection is just to slow and you get too much false positive/delay.

So this is a frustration topic but... I am not that frustrated.

The NFC Ring team took a very professionnal decision inline with all of their other decision : ship a complete product with good quality instead of a incomplete product to please the masses. Then additionnally they offered a not even required compensation for people that ordered carbon fiber.

I respect this desision.

My idea is to use the NFC ring to wake the device and light the screen of the phone when it detect the ring.

This is not a new idead. It has been featured on Xda developpers. Here is the original link.
I can't post the link because of forum rules, but search for the news "Enable NFC While Locked or Screen Off on the Galaxy S III" on XDA Developpers.

The the user want to access it's phone right now, it need to press the power button, and the nfc ring team released a software to automatically unlock the phone.
Well... you can do better. When the user take the phone in it's hand.... what else could that be for if not to use the phone ?
I suggest to modify the operating software to wake the device as soon as my hand reach it. It would also unlock the display. This means that I could instantly (or almost) use my device without any hassle of button pressing/unlocking.

I am not a developpers, but I know that this need custom modification to be done. I see that there are some custom version of the nfc program in the operating system, but I guess someone could make a Xposed Module (that would be available to every rooted phone, regardless of the brand) and enable to scan nfc tag when the screen if off.
I do not know about what would be the effect on battery usage. I am currently trying to do something similar with tasker and the proximity sensor (if my phone was in my pocket and I remove it from there, wake the device .. king of logic).