Gamma Correction and Shadow Detail bit allocation

I’m having slight difficulties understanding how a gamma correction increases details in the shadows(where our eyes are more sensitive). Once the bits have been been reallocated to the shadows after applying an INVERSE GAMMA/GAMMA CORRECTION in the camera wouldn’t all that detail just be lost as the monitor would apply a GAMMA to counter the inverse gamma thus bringing the image luminance back to a linear function. Or are the code values saved after gamma correction and only the brightness is brought down.

I’ll use an example I took from the video "Diving into dynamic range" from Filmmaker IQ on youtube.

From what I understood, in his example he uses an 8 STOP(the triangles represent the stops) camera with an 8 bit depth(Not exactly sure about this I got abit confused here. Please correct me If I’m wrong)

Basically once the GAMMA of the screen is applied to the above OETF GAMMA curve and it goes back to the LINEAR one why wouldn’t we loose all the details in the shadow again??

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Here’s the link to the video just in case: https://www.youtube.com/watch?v=2sshGdMgJxQ

Perhaps I didn’t explain my problem clearly allow me to retry from the start.

So, whenever we have a purely linear function we lose too much detail in the shadows as the majority of the values are assigned to the highlights(which we as humans can’t discern as well). As shown in(1st image below)

Therefore this is why the camera applies an inverse gamma(e.g ^1/2.2)which allows more values to be reallocated to the shadows. As shown in (2nd image below)

Now my question is once this OETF Gamma is fed to the monitor, wouldn’t the display just apply its own Gamma(e.g ^2.2) to the OETF curve and thus make it linear and bring us back to step 1 where we have lost all our values in the shadows again?? Or would it as Rawshooter say "Save the coded values" and we’d end up with something like below(3rd image below)

I’m basically asking that once the coded values for the tonal shades are redistributed in a gamma encoded curve do they stay that way even though the gamma encoded curve will be decoded to display a linear function in the final image(4th image below)