With the benefit of hindsight, the freight route would have been of value today, and perhaps domestic intermodal terminals could have been located along it instead of the WCML and branches. I'm less sure about the passenger service at the north end, as the Robin Hood Line sort of serves that purpose and could be connected to bits of the GC or other lines revived, if the will and the money was there for better links to the former coalfield. As already mentioned, the Midland was much better connected, mainly because it got there several decades sooner. The Victoria site is only close to the centre of activity in Nottingham today because of the shopping centre that replaced the station - if it had stayed open it would have been no better than Midland in this respect.
As for Leicester-Rugby, a corridor between the East Midlands and Oxford or beyond is something of a missing link. In the modern day this would ideally run via Northampton, MK and Bletchley instead of missing the major population centres as the GC did. At the Leicester end the Midland already connected and the GC could have connected to the Nuneaton line. At Rugby either the GC or the Midland could probably have terminated in their own platforms on the east side of the main station, but both would have been difficult to connect for through passenger services towards Northampton, thence Bletchley and Oxford. The Midland could have been less difficult, because at least it entered Rugby going in the right direction.
To re-use the GC as a high speed main line:
You need to go towards Birmingham so discard the bit north of where it turns eastwards.
The London end has commuter trains getting in the way, and Marylebone is too small to be the London terminus, so discard that bit too and use the GC-GW joint line instead. But there's no room in Paddington either.
You can't run close to homes, so go round or under any built-up areas.
You're left with the section between Brackley and Aylesbury exclusive. South of there it needs to go further west, why not add an interchange at, say, Old Oak Common, and head for Euston where there's a bit more space?