The key link that has been lost with closure of the GC main line is from Leicester/East Midlands to Banbury (as a gateway to Southern England), which bypassed Birmingham/West Midlands and was more direct. The fact that the line passed through Rugby is of lesser consequence. This section could have been retained by connecting the GC main line to the ex-Midland Rugby-Leicester line south of Countesthorpe.
This is true, but the traffic that went that way (also Woodford Halse - Fenny Compton - Stratford and onwards) was mainly minerals, which died; and much of the reason for doing it was because it was necessary to care about which company or set of companies owned the lines any individual flow was going over rather than just selecting the optimum route on its own merits. It was a link between the industrial areas of the Midland axis and South Wales, and a lot of its importance was as an element in all the diddling on that went on because the GWR had South Wales more or less stitched up and everyone else kept trying to find ways in the back door.
That reason of course no longer applied under nationalisation, and it was probably reasonable to expect the iron ore traffic to dry up seeing how foreign sources had been taking over from domestic ones for a long time already. It probably was
not reasonable to expect massive loss of coal traffic, since the stuff was vital and we still reckoned to have 300 years or so of it left at the time (Thatcher vs Scargill knocked an order of magnitude off that estimate overnight); but it would have been reasonable to expect big changes in its routing since that was starting to happen already.
As part of a long distance passenger route between the Midland axis and the South Coast avoiding Birmingham (which is what I think you're mainly talking about) I'm not all that convinced, because Leicester-Banbury via Nuneaton-Coventry-Leamington is only a little bit further round; most of the origins/destinations are sufficiently far north or south of the area in question that even going via Birmingham itself is not, proportionately, all
that much further round; and going via Birmingham has a good deal to be said in favour of it for all the connections it gives. Going through Rugby is really the main thing that route
does have going for it - but most long distance services would still want to go via Birmingham anyway, so the service Rugby would get would be rather crap.
As it happens I do favour the idea of serving Rugby on that axis, but I can't see my way to giving it a very high priority, nor to trying for anything more than a local service. It would be possible to make a big difference to the attractiveness of commuting to Leicester because the alternative is twice as far round with a change in the middle, but that's about all, and the proportionate gain for anything more than that drops off very rapidly.
The lack of any significant cross-country links between the main lines to the north, from London as far north as the Leamington Spa-Coventry-Nuneaton-Leicester-Peterborough axis, has been an unfortunate consequence of the 1960s closures.
I agree. It's a pain in the arse, and it's a general failing, not just a railway-specific one - transverse journeys by road tend to turn into a bit of a trek as well, though at least they aren't impossible. I look with favour on ideas relating to filling the holes back in; the thing is here that while this hole is directly relevant to the subject of the discussion, it's not one of the more important or severe ones overall.
Would such a tight curve (Rugby Station facing south-Clifton Mill-GC South line to loop over the Birdcage Bridge), between two junctions on a gradient, been really practicable? For heavy freight traffic? Seems unlikely. Maybe on the DLR or in some colliery sidings.
The radius of the new curve would not need to be as tight as (to pick some apposite examples) the minimum on the eastern tendril of the Market Harborough line connection, or the north to north-west curve towards Fenny Compton at Woodford Halse (which is the same), and certainly the latter saw a lot of freight. The 95 metre contour is at track level for both the GC north of the Avon and the MH line at Clifton Mill, so although you would need an embankment between those two points the line on top of it would be level. And the GC seems to have planned something of the sort, although they didn't do it in the end. In effect I'm taking more or less the same position as the GC did: I think it would have been practical, but the question is whether it's worth doing it at all, which I'm not necessarily as convinced of as I suppose it might seem.
The congested bits of both the MML and WCML are the bits further south than that - effectively Bedford - London and Milton Keynes - London respectively.
Yes, but Leicester is the first point at which you can connect the GC to the MML with just a bit of chord. It also makes possible a rather less awkward connection pointing towards Birmingham than you could get at Rugby (another reason to consider that although the connection at Rugby would be possible, that doesn't necessarily mean it's worth the bother).
The GCR was a 2 track mainline so would actually be worse for freight than either the MML or WCML are, both of which are 4 track over those distances.
...and as you admit, those 4 tracks are nevertheless congested, because they have lots of passenger services of different kinds using them as well as freight. So if you take the freight off them and run it up the GC instead (which
was a freight mainline) the whole tangle gets much simpler to sort out and all the services are better off.
Equally HS2 is designed to free up the paths on the southern WCML and possibly even the MML as well.
Oh, aye, and it gives you pie in the sky and fairies at the bottom of your garden and 10tph additional capacity on every line anywhere between London and Scotland. Amazing what you can do merely by shifting all those non-stop London-Birmingham services onto an entire brand new line that costs such a hideous and ever-increasing amount of money that even the government who love it so much classify it as "probably not gonna work". HS2 is this forum's blind spot where all the usual emphasis on realism and practicality undergoes a bizarre inversion and magical thinking becomes the order of the day, and I am disinclined to further annoy myself by talking about it.
I'm talking about the Great Central, and saying that we could have made the extra capacity available through shifting the freight off the route any time from half a century ago onwards, simply by not destroying what we already had, plus a couple of km or so of new chordage.
And sending freight all the way to Leicester (from the south) to access Birmingham is extending the journey quite a bit - have you looked at a map ?
Lots of 'em

Yes, it is further round, but not by all that much in proportion to the whole run. And it's the most straightforward way to achieve a good connection, that doesn't interfere with the WCML at all, by means of minimal additions to what was already there. There are plenty of other examples of freight routings that seem excessively circuitous because they are operationally easier that way.