Which is all well and good, but the difference in practice (within the blocks) isn't huge.
Classification issues aside, I think the example you've given above doesn't put Reith in the best light.
Firstly, they're far from the best renderings possible. Shrunk down by the forum makes it worse.
Secondly, the weight chosen for NEWS is far too heavy, making the BBC lettering look anaemic next to it.
Thirdly, the unstacked versions of the logos have the lettering too large in both examples- doing as is (or should be) done now, and having the height of the text the same as the letters in the blocks looks far better, I find.
The BBC lettering was customised in weight (and shape, I recall), so I would suggest that the same may well apply in this case, were it to ever happen.
I made that image when the BBC Sport rebrand was announced. But either way, just using Reith in the blocks will not work without re-drawing the letter shapes. In my opinion, its not worth the effort to change it. The Logo is seen as a self contained item, no longer striped with text to create logos.
Also Eric Gill sculpted parts of Broadcasting House, so keeping the Gill Sans based lettering is a tie to the history too.
I am not keen on the BBC News logo using Reith in the way I demonstrated, but I was just using the Sport logo as a guide to demonstrate how it could look.