No you don't. As the Crow flies, the most direct route to Manchester is via Nuneaton (as per the current WCML) and then via the Churnet Valley (which you have to avoid). We are only talking about following the M1 south of Rugby.
No, the straight line between central London and central Manchester passes through Northampton, just west of Leicester and Derby and through the Peak District roughly between Ashbourne and Marple, passing just west of Buxton. The old Midland route was more direct than the WCML.
When you mentioned the M1 corridor in your previous comment, you mentioned it being the most direct route to Sheffield and Leeds, that doesn't give me the impression that you just meant to use it south of Rugby.
That makes no sense. Why would a non-stop train from London to Manchester be less busy if it went via the M1 corridor rather than the Chilterns? It would be shorter and quicker...
If HS2 just joined London and Manchester, maybe with an awkward branch to Birmingham, it would be far less busy than if it served both Birmingham and Manchester properly. The MML already carries the vast majority of of passengers between Sheffield and London, yet it's the quietest of the north-of-London main lines.
How do you work that out? A new line south of Rugby that takes existing traffic off won't ease capacity. If you build a London - Rugby via MK line, you remove ALL existing services of the existing line and still keep all the other stations served (except Watford Junction).
Is this another new line in addition to HS2? Following the WCML closely enough to serve MK and Rugby isn't practical. By "ALL existing services", I assume you mean all Intercity services? If not, how do you intend to serve Hemel Hempstead, Leighton Buzzard, Tring, etc?
If you build a direct line between London and Manchester, you take at most 3tph in each direction off the WCML (less, because stopping services will be needed to serve Stoke, Stafford, Rugby, etc.)
The Coventry corridor needs 4-tracking end of. HS2 will be used as an excuse not to but it won't help provide the extra paths to allow fast trains to Coventry & Northampton aswell as the local services.
I agree that the Coventry corridor needs 4-tracking. However, there isn't room. Acquiring all the land and modifying all the structures to do it is likely to cost somewhere in the billions, at which point you've already spent a significant proportion of the money for HS2, which will reduce the number of "fast" services being held up by the stoppers and the need for as many New St.-International trains, helping with the capacity issues.