All you do is increase the inefficiencies and create pinch points in different places. You seem unable to comprehend why that's a problem.
And if Bakewell (popn 4,000) with limited demand for travel to Derby or Buxton is your best justification for this line, then you really don't have a justification.
Manchester - Derby can be done just as quickly via Stoke with 1 change. Leicester - Manchester doesn't really have any demand as the Project Rio services demonstrated.
So then you're left with freight. Hope Valley upgrades can address some of it. And if a new freight line is needed, the find a decent formation, not a rehash of a suboptimal Victorian one with gradients, curves which goes through tourist areas.
How? as I noted earlier, its all very well taking it in isolation, but do all the extra trains fit at either end, where are they going etc.. how does this make those parts of the railway more efficient?
I would divert the Manchester - Nottingham service via Derby which would bring an extra large town (Derby) into the catchment and extend the Buxton stopper to join with the Matlock stopper to provide the local service through the peak.
This automatically frees up two paths in and out of Sheffield to improve services/provide extra capacity along the Hope Valley and Chesterfield lines - You only have to find one extra at the Derby and Manchester end.
Added to this, you have an extra route to get freight out of the way at Peak forest.
I'd remind people that some years ago we didn't run passenger trains along copy pit, most going between West Yorks and South Lancashire had to go via Manchester.
Now we do send trains over Copy Pit. The pinch points are still there in Manchester, Preston and West Yorkshire. If anything they're busier than ever. Would anyone seriously suggest that not having the direct trains between West Yorkshire and Preston would be an operational improvement ?
Can you put some numbers on the services you anticipate on the Hope Valley and the High Peak routes after reinstatement? It's all very well declaring that it will reduce the inefficiencies, but skilled people have looked at this multiple times and said it wouldn't, so obviously it's not readily apparent and they've missed something that you've seen.
No, skilled people i.e. train planners look at what they can get out of the existing network. They haven't got time to work out the benefits of the non-existant route.