More generally, surely the point of the Thames-Clyde 'Express' for most of its life was that it facilitated inter-regional journeys, hence the importance of connections at placers like Kettering or Trent, or convoluted routings via Nottingham and/or Derby, etc.
If you wanted to travel rapidly between London and Glasgow you would use the WCML.
I realise that in an earlier era the Midland Railway might have wanted to create the idea of a prestigious train but those days vanished many years ago.
I suspect it is true that by the 1930s most people would choose the WCML if travelling from London, but I think you will find that up to 1962-ish the Midland route was still advertised/marketed. as a 'standard' route to Scotland. Surely someone can check, but ITYWF Thames-Clyde, for example, only began to introduce stops on the S&C (Appleby, Settle) from about 64 or 65.
I can only assume this practice continued so late in the day because there was no strong ovearall management, and so the lines of management inherited in the 1922 grouping just carried on into the early 60s, and of course, each manager wanted to keep his 'flagship' services. If you think about it, it's very similar to the LMS keeping the St Pancras - Manchester Central route as a front-line service when modern management thought would have said immediately "Let's cull those trains immediately and concentrate on running a fast, hourly service Euston - Man London Rd."
But they didn't do that. Of course, come WCML electification, the Midland route was more or less ready to act as a back-up service immediately as a result.
Yes - some scheduled passenger services did use the Chaddesden route. Travelled on one myself into Derby, but would have to search old records to find out which train it was. (Might have been a Skegness to Derby services, to avoid having to reverse the Class B1 before sending it back home to the Eastern Region)
The Chaddesden route was used a bit more than that at times. ITYWF from summer 1960 for two years or so the approx 12.15 Manchester LR - Euston was diverted Stoke - Derby - St Pancras and in the down a 13.55 St Pancras - Derby - Stoke - Manchester London Rd both used the Chadd route for at least some of this time. There was a photo of a Class 40 (the down train was almost always a Camden Cl 40) arriving in Derby from the north in a TI (or Railway Magazine, not sure) soon after these diversions were introduced to relieve the WCML of traffic.
The Chadd route was also used for some Nottingham - Birmingham - (possibly Bristol or Cardiff) trains too, I believe, but I'm not sure if these were regular or summer specials.