LNER are running 1TPH less out of Kings Cross than they'd planned for after the now delayed recast. Avanti are running 3TPH less out of Euston than their pre covid timetable never mind what their timetable would have been after any now delayed recasts. So under that princple Avanti should be doing better than LNER for crew not worse.
It's both about the crewing position as well as the timetable position. Avanti have long been reliant on a lot of RDW, LNER less so.
Not all tph are equal - 1tph extra to Newcastle takes a lot more in terms of resources than 1tph to Birmingham, as the journey time (which influences the numbers of units and hence crew required) is twice as long.
Southeastern are in many areas running what they ran pre covid.
In some areas, yes. In others they are only running half of pre-Covid. But they were generally not quite in as parlous a situation as Avanti, TPE etc. pre-Covid. And they are benefitting from having to run fewer peak extras.
TPE are running around 50% of their pre covid timetable on both the Newcastle - Manchester/Liverpool
Yes and no - whilst it's nominally half the number of tph, halving the service doesn't mean that you need half the number of units or traincrew. The ratio of services to traincrew isn't at all linear, so you often have to cut a disproportionate number of services (say 50%) to save a smaller number of crew diagrams (say 25%).
and Scotland - Manchester routes
That route is (in theory) running to its full pre-Covid frequency, albeit with a few early and late trains withdrawn.
, whilst most Scarborough services are only running as shuttles to/from York, and TPE are still suffering mass cancellations. Avanti are running 2/3 of their pre covid timetable to Manchester, 2/3 of their pre covid timetable to Birmingham, and 75% of their pre covid timetable north of Preston, and again that service pattern is suffering mass cancellations.
This I'd agree with. But unfortunately the crewing position is the more dominant factor.
You only have to take a look at a large number of what-were franchises; West Coast, East Coast, South Western, East Midlands and ScotRail to name a few, to see drastic changes in service quality very quickly after a change of ownership without any (or with minimal) changes to the timetables.
I'm not sure I recognise all of those as examples, but in many cases it is down to the contents of the bid and any cost cutting that was required to win it. So in essence - it isn't a matter of the particular owning group in charge, but rather the conditions of the bid they had to make to win it. And if they hadn't won it, someone else would have done on largely similar terms.
EMR stable their units for the early and late Liverpool services in Nottingham. Whilst I'm not suggesting everything that uses Edge Hill go as far afield as that as an alternative, it shows that alternatives are available and they don't neccessarily have to be in Liverpool itself.
Of course it's possible to come up with alternatives in due course, but they can't be put into place overnight. EMR (and their predecessors) have stabled 158s in Nottingham for the last 3 decades - it's a well established way of doing things.
Meanwhile good luck finding space for 5 extra Pendolinos somewhere near Liverpool... You'd have to build a new depot or expand an existing one. It would be terribly inefficient from a crewing perspective. And all for what - to avoid paying Alstom a little bit more for 24/7 opening hours? I don't think so.
Contracts often have break clauses and can in the vast majority of cases be terminated early under set circumstances. Not providing the contracted number of working units for service each day, which is what the result of units being stuck at Edge Hill is, will be in breach of the contract unless the contract is exceptionally poorly written.
I admire your optimism

. There are generally breach thresholds and set procedures for terminating the contract; one set being unavailable won't come remotely close to triggering that.
In the case of TPE, Alstom merely service the units and aren't responsible for any faults that develop. They are doing exactly what the contract requires them to do.
What bugged me last Thursday was that due to a TPE cancellation from Glasgow, Avanti generously picked up all their passengers, and by where I boarded at Carlisle the train was already packed. The seat reservation system wasn't working (it showed reservations but not the correct ones - someone was in my reserved seat and when the display is wrong you can hardly ask them to move), and I had to vacate a seat I'd managed to find when someone appeared with a valid reservation slip, and it was altogether unpleasant. I had to walk through from A to E before I found a seat that wasn't occupied either by a person or someone's luggage (the racks and overheads were full), and ended up being harassed by a chap at the window seat who insisted I swapped and took the window seat as he was getting out before me, then harangued me because although I was polite at first, I wasn't interested in his bombardment of questions and eventually told him I would appreciate it if he stopped cross-examining me. He ended up being quite offensive, luckily a chap in the row in front turned round and gave me a bit of support.
I would have claimed your reserved seat unless the TM had announced that reservations were being lifted. Though it's unfortunate for the person who sat in a seat they thought was unreserved, ultimately if you don't have a reservation, you're not guaranteed a seat.
I obviously don't think the TPE customers should have been stranded in Glasgow, but if there was a later TPE service perhaps they should have been advised to wait for that. Oh and the interminable station announcements about Avanti "this train is expected to be very busy, so if you have a flexible ticket you might wish to wait for another service for a more comfortable journey" - who the heck is going to do that, given the announcement seems to apply to every service? Especially when they are an hour apart. If anyone has actually ever made the decision to do that I would love to know. My money is on it being a waste of air time, and so annoying.
It depends on exactly which service was cancelled (was it a Liverpool service or a Manchester service an hour before a Liverpool one?), but generally the frequency of TPE services from Glasgow is every 2 hours. So passengers would in all likelihood have had the right to be re-routed at the earliest opportunity.
Accordingly TPE didn't exactly have a choice about re-routing passengers onto the next Avanti service.
And as it happens, there is standing ticket acceptance in place on Avanti in the event of TPE cancellations.