PART 1 - Theoretical Northbound paths
So, firstly we have to think about paths north from Kings Cross as including every train passing Alexandra Palace to Finsbury Park. This includes Thameslink through the core and GN from Kings Cross and from Moorgate.
Next let's think of constraints. Anything coming off the Moorgate doesn't occupy any constraints northbound, now that the new Stevenage platform is built. We can discount these.
EDIT: see posts 10, 11 & 13 below for exact reasoning why these trains can be discounted.
However, anything running main line past Welwyn Garden City northbound has to cross the two track section at Digswell. A fast train will take approximately a minute to clear this on greens. A train that has come off the slows, crosses this section and then rejoins the slows will take approximately three minutes. That three minutes is equivalent to one path out of Kings Cross, so for every train that does this, that's one fast path gone.*
Now, paths themselves.
Thameslink has 2tph semi-fast from Kings Cross to Cambridge. These go slow line and depart at xx.22 and xx.52, but cross the two track section at Digswell at xx.02 and xx.32. A fast train takes 17 minutes to reach this section, so that's the paths at xx.15 and xx.45 gone.
The four Thameslink trains out of the core run fast line as far as Hitchin/Stevenage. The Brighton to Cambridge trains occupy the equivalent of xx.09 and xx.39 paths. As JonathanH pointed out, the xx.06 and xx.36 paths are occupied by LNER stops at Stevenage, so those don't exist. The Peterborough to Horsham trains also run fast as far as Stevenage. These occupy the xx.21 and xx.51 paths.
The last ones are the Cambridge Fliers, occupying the fast lines non stop as far at Hitchin. These occupy the xx.12 and xx.42 paths.
So we have, theoretically a path every 3 minutes leaving Kings Cross, but as we've seen, not all of these are practical as a lot of them are used by TL and GN.
*It's actually a little more for trains that call at Welwyn North, but we can ignore that rounding error.
PART 2 - ECML capacity constraints
So, as outlined above, a lot of the paths are used, they're just not used by trains at LNER.
Thameslink use six paths per hour, four fast and two on the slows that stop at Welwyn North. LNER use five, although this changes in some hours, and itself will grow to six after Werrington/Kings Cross Remodelling (Dec 2022 timetable change). Great Northern use two for the Cambridge fliers. The OA operators use 1 between them on average. Add in two for resilience and you only get 16.
But a train can leave every three minutes. Add that up and that's 20 paths per hour? We're missing four.
Obviously this sometimes doesn't work out, because trains turn off at Hitchin or terminate at Peterborough, and therefore those paths aren't used beyond there, but in practice, these paths are used up further along the route.
1. The flat crossing at Newark uses 1 or 2 paths per crossing move, and there are two paths per hour, in each direction. Sometimes these can be parallel, but that doesn't always work out.
2. Freight trains using up 2 Northbound paths and 1 Southbound path while crossing over at Werrington Jn. Obviously this is being fixed, but it was for a while the major capacity constraint on the ECML. There are two freight paths northbound per hour.
Equally, southbound freight trains from Sleaford have to cross the northbound ECML as well, although they usually hold them outside Peterborough to wait for a quiet time to cross. This again uses another northbound path, but can be parallel with northbound freight.
Passenger trains from Sleaford now use the reversible slow line, and terminate at Peterborough, not using capacity.
3. East Midlands Railway run a train in each direction every hour (at 90mph Vs 125mph) from Peterborough to Grantham. This uses some paths.
4. There's a lot of freight on the ECML, and that's all max 75mph. It takes a long time to accelerate, decelerate or loop a 375m freight train, and that's a lot of paths. Granted, a lot of it gets diverted off, but what's there still has to travel across the same two track sections as the passenger trains.
5. Platform capacity at Kings Cross was lacking. Additionally, every fast train calling at Grantham or Newark or Durham among others eats a path, because you're stopping a train on a two track railway, and therefore stopping all trains behind it.
All paths south of Leeds are used up, at least as far as Wakefield and the flat junctions towards Sheffield. No more trains can be run to Leeds without impacting services there. The same applies for services to Newcastle, where the two track section to Darlington and the flat junction at Northallerton eats up capacity.
6. This brings us to our final issue, there's nowhere to send them. Anywhere worthwhile hasn't got the capacity, there wasn't a need for Lincoln to have an hourly service, and no real desire to terminate more trains at York. There weren't the spare trains, the slack in the timetable wasn't there, and noone really needed to. Indeed it was only really a decade ago that the desire to get new paths was set forth, and schemes such as King's Cross remodelling and Werrington will help realise these new paths.
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There are a lot more complex reasons, and I haven't really fine into too much depth about the actual pathing out of Kings Cross, but I hope this gives an overview of the complexity of planning for more trains along such a busy and complex route. I apologise of it's a bit hard to follow, but I would add though, that the WCML is much, much worse.