The Elizabeth line (it was called Cross Rail during development) is the first train line that combines both the national rail infrastructure and the London Underground. Combining the signalling systems has be an incredible complex problem. There is quite a bit online to read about it:
https://www.wired.co.uk/article/crossrail-delay-logistics
https://www.crossrail.co.uk/project/our-plan-to-complete-the...
No, it isn't. Historically the boundary between the Underground and the rest of the "mainline" railway network used to be even fuzzier with lots of through-running, but even today a few stretches of lines served by both Underground and mainline trains remain (the Metropolitan Line north of Harrow-on-the-Hill, the outer ends of the District Line to both Richmond and Wimbledon, the Bakerloo Line north of Queen's Park).
What's true though is that due to its capacity requirements, Crossrail is using a different signalling system than the rest of the railway network, which means lovely interface issues at the transition points…
So the trains will need to seamlessly transition between TPWS on the national railway lines, then CBTC in the London underground, and ETCS when going to Heathrow airport.
I think this is what they call integration hell, and is responsible for delays of several years. When you consider the complexity, but also the safety-critical nature of these train signalling systems, it's no wonder it has overrun.
So the "integration hell" is trying to integrate between services with completely different service strategies.
Both have planned where every trains should be at any moment. The difference is the amount of leeway you have to fix operational issues.
You can think of the metro as a “timetabled” system where the density is so high you have no margin. When something happen you have to shift and cancel. Train lines start having exactly the same issues when they operate near capacity.
Both systems have the same constraints. Trains can’t overtake. You can only have one per block. The integration difficulty is that they don’t manage blocks in the same way and use different signalling. You see the same issue on lines which are shared between SNCF and RATP in Paris. The A line used to require a driver change at the junction.
There actually is a timetable for all of London Underground, it's needed to roster drivers. But the only services ordinary passengers actually care about are the first ones and the last ones, Amersham to Baker Street last train is 23:22, if you're out of the city too late for that, you're not getting a tube home. When/ where there is Night Tube you probably don't care, as most people taking Night Tube aren't exactly on a tight schedule.
However, the Underground is not run at capacity as you imply. There is deliberately wiggle room for "recovery" from events even at 30tph. This is one reason why the Circle Line isn't a circle any more, it makes it possible to recover more easily on that line.
Part of the underground is definitely run so close to that you can think of it as run at capacity. There is some wiggle room especially where it’s manually driven and during the slowest hours but nothing comparable to a typical train service. I am admittedly a lot more familiar with the French situation - I worked for the French national infrastructure manager on modernising traffic management and for TGV on their planning system - but some lines in Paris the block sec is so low even the slightest delay in doors closing will lead to cancellation and trains can’t be controlled manually.
EDIT: I guess this is the literal definition of "Civil Engineer"
(I know it’s a forum post but that old website is a wealth of information about the District Line).