The best use for the GCR main line would have been as an alternative route for North/South freight. With a few new/modified connections, it could have carried much of the inter-modal freight traffic to & from Trafford Park and Yorkshire, releasing paths on much of the WCML (southern end) for more passenger services.
Passenger services to London Marylebone could never have competed with MML, but it might have become a useful Cross Country route avoiding Birmingham, for the East Midlands (Nottingham / Leicester) to Oxford / Reading / South Coast via Woodford Halse & Banbury. Whilst closing some of the less busy, there could have been a potentially useful semi-fast / commuting service between Sheffield and Nottingham or Leicester serving places like Staveley, Hucknall & Bulwell.
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.
Actually the Beeching Report did propose to close much of the Midland Mainline, many of the maps show it closed entirely south of Leicester, with Leicester connecting to the WCML via Nuneaton. And of course St Pancras was planned for demolition.
As to why the GCR London extension was built, well it is for much the same reason as we are building HS2 today, it wasn't intended to stop at every town along the route, but to the provide fast, direct services between London and the North. And just as HS2 doesn't stop at every town along the route because it wouldn't make sense, neither did GCR.
While I can understand the logic of closing the GCR I think it was unfortunate that there wasn't more effort made to retain some of the good parts, after all the UK rail network is a patchwork quilt sewn together from different bits from different companies across the years. The GCR did have some good assets which should have been kept, particularly in Nottingham with Victoria station right in the city centre and the north-south route straight through through the city, this could have allowed London to Sheffield services through Nottingham to be retained which would have benefitted the city's connectivity in the long term, much better than the Midland station is was left with.
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?