It's not true to say that the GC didn't serve anywhere. It did. It served lots of places. The thing is that someone else was already serving them too.
The GC was very like a sort of slightly further north version of the Midland; both of them developed as provincial outfits connecting up the different bits of northern industrial areas, both of them were heavily concerned with major coal mining areas (often the same ones, side by side), and both of them found as a natural result that dragging endless tons of the stuff down to this giant pollution hotspot in the south where a significant chunk of the country's population all sat round on top of each other burning it like there was no tomorrow became a similarly important concern. And both of them accordingly went through the same kind of process of trying to find ways to do that without spending too much money by trying to batten on to existing lines for the southern end of the run, both of them found that basically didn't work because the other lines were more concerned with their own doings, and both of them eventually went "sod it, we'll have to build our own route to London", and did build it despite some lack of internal unanimity over the idea. It just took the GC a bit longer to get pushed over that edge.
There was a tremendous amount of overlap between the operations of the two, with a great many sources and destinations served by both of them over their own independent routes, even when those independent routes ran together for many miles down the same bit of valley twisting over and around each other like mating snakes (and the GN making a threesome in some places). Similarly they eventually worked out, after trying both options a few times, that it was more productive to cooperate rather than bicker when it came to making progress against the LNWR's efforts to keep Manchester and the Mersey to itself, and ended up spawning this kind of shared subsidiary outfit to cover that area together.
It would have made a lot of sense for them to have settled on cooperation both earlier on and to a greater degree, and simply merged together, some time before the Midland route to London was built. (To be sure this would have meant overcoming not only internal tribalism but also the dimwit politicians worshipping the demon of competition, who aren't a new problem by any means.) Once the two systems finally did become forced into the charge of one outfit, the profusion of duplicate routes they had come up with due to not having acted together became too obviously silly to ignore, and getting rid of the crappier bits was only sensible. Naturally this affected the GC routes more because they were usually the second or third arrivals and had to squeeze their lines through what was left after someone else had taken the best alignments.
So the loss of most of it north of Leicester was really only to be expected once people started being a little bit sensible. Unfortunately they were not sensible enough not to go too far and we lost most of the bits that were not more or less useless alongside the bits that were.
Between Leicester and Rugby there is a silly great hole which forms part of the severe anisotropy that afflicts transport in England generally (road as well as rail, although rail is notably worse). Two more or less parallel defunct rail routes used to cross this hole. Me, I'd probably have kept the Midland route as the exit from Leicester, joined it to the GC where the two cross over, and keep a stub of GC north from that new junction plus a couple of km of new chord to make a westward connection with the Nuneaton line. ("You can't do that!" Oh yes I can, the land's not built on and they shoved a whole flipping motorway through nearly the same spot.) Then at the south end, I would have at least added a north to east chord at Harlesden off the Dudding Hill line, and perhaps also carried on the programme of four-tracking the Met through Ricky to Amersham; and of course we have the GC/GW joint route which was always the main access once it opened. That gives you feeds both from the NLL and from the GWR and LSWR routes, and provides a freight route bypassing both the southern end of the MML on the way to the East Midlands etc, and via the Nuneaton flyover also bypassing the southern end of the WCML for freight to Birmingham and beyond.
As for the silly hole, it would also be straightforward to add a chord north of the WCML at Rugby (again, across unbuilt land) cutting across from the GC heading south onto the Market Harborough line heading south-west, which would allow local passenger services between Leicester and the WCML station at Rugby. You might even be able to squeeze in a curve by which services having reversed at Rugby could get back onto the GC heading south, as long as you didn't try to run Pacers round it. You could also add a chord where the two lines cross at Brackley to allow north-south passenger locals to call at both Brackley and Buckingham, if you kept Buckingham as well, which I would because Buckingham not having a railway is crap. Though I will admit there's an increasing amount of something akin to scraping the barrel coming in here.
I'm not entirely convinced about the ideas proposed around keeping Victoria station in Nottingham instead of Midland, probably because I don't know Nottingham well enough to understand the advantages, although I do agree that Midland is a bit stuck out on the edge. Assuming it would be useful, though, then from checking out six-inch 1947 OS maps and guessing that not much more stuff had got built by the sixties, it actually isn't too hard. A connection to the routes out of the east end of Midland is there already, and sorting its deficiencies is basically a matter of rearranging some of the use of what is railway land already. Connecting the Victoria route north to the Midland to Mansfield is trivial, and connecting it northwards onto the Earwash Valley line only needs a chord. The only notably awkward bit is connecting to the route to Trent and onwards, but it could still be done by looping round and hitting the Lenton triangle from the south side.
On a different aspect, there does seem to be a bit of false dichotomy going on in the "keep vs close" arguments, in that much of the motivation for "keep" seems to be based on the only alternative being "close and destroy" so that to avoid that calamity it is necessary to argue for "keep it intact and running" however desperate the cause may be. But there is in fact a third class of options based around "close" being interpreted as nothing more than "just stop using it", which is the way it should have been interpreted (and not only for the GC).