I agree that many of these traffic flows are poorly served, but I don't think that the Aldwych Branch is really useful in any way to serve them.
For Clerkenwell, the easiest and cheapest way to serve it would surely be to add a new station on the Circle line, somewhere around Mount Pleasant post office - I'm afraid I don't know the area well, but this looks about the furthest away from the existing stations
If you did want a new tube line from Waterloo to Clerkenwell, the direct route (to Mount Pleasant PO again), the direct route is 2.5km. It goes past Chancery Lane, so an interchange with the central line is quite practical. Going via Aldwych and Holburn increases the distance to 2.77km, of which .68km is the existing tunnels from Aldwych to Holburn, so the saving of tunnelling distance is only 400m on , however this would be offset by the need to build two separate tunnels from each end, rather than going straight through, plus I strongly suspect the cost of modifications to Aldwych and Holburn stations would be more than the cost of building from scratch. As you have mentionned, platform lengths are an issue - I really don't see that building a brand new line that can only cope with short trains is a real option, plus Holburn itself would need complete remodelling, both track-wise, as the existing dead-end platform, and one with connections to the Piccadilly mainline would need massive modifications to make them instead connect with tunnels to the north, and passenger wise, as the existing station barely copes with the traffic from 2 lines, it certainly couldn't cope with a 3rd. Aldwych would also need a rebuild to make it into a through station, and to add escalators and new elevators to the surface. Finally, I would doubt we'll see another narrow-diameter tube line across London - the current trend for mainline-sized tunnels is probably much more likely to continue, as the capacity is so much greater and the cost not much greater with modern tunnelling methods.
Most of your other (very reasonable) desires are provided for by Crossrail 2, which provides the additional capacity into Euston, and also frees up space at Clapham junction for more semi-fast stoppers, so I don't think that the Adwych branch is the right solution to any of these problems.