VTEC also run one train a day between Aberdeen and Leeds via Edinburgh, so their trains could go that way more often.
No, they really couldn't/shouldn't. VTEC's one return service between Leeds and Aberdeen operates for logistical reasons, not specifically to provide a service on the Leeds to York corridor. VTEC's main business is travel to and from London on the ECML proper. Diverting trains via Leeds would significantly slow things down.
The Edinburgh-Newcastle-Leeds corridor is perfectly well served by XC and TPE. They are the most sensible operators to do so.
Glasgow-Sheffield is a lot faster via the WCML, even without any direct trains, as going via Edinburgh requires a massive detour. Leeds-Glasgow isn't as far via Manchester, either, but the line speeds and lack of direct trains make it slower that way.
It's not a lot faster, it's not more than half an hour in many cases. It is indeed sometimes the fastest option to travel on the direct train (for example, the 06:01 departure from GLC). For many people, the benefits of a direct train could outweigh the relatively modest time savings afforded by the inconvenience of changing. Even beyond Sheffield there is still a benefit in the direct train - for people going to Derby, it's only 20 minutes longer on the direct train and avoids the hassle of three changes in some cases.
Of course, for Birmingham and beyond the benefits of the direct trains are vastly outweighed by the extra time required (up to two hours), but until then it's a relatively modest time penalty for a significantly more convenient journey, which would be useful for quite a few people (elderly people, those with luggage, those who are not familiar with traveling by train...).
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Would Leeds to Glasgow via Preston be quickest? And Sheffield perhaps via Warrington and a quick walk? Both trying to grab the London trains of course.
There's very little in it. The journey planners again suggest no more than 20 minutes difference, although for Leeds the direct train via the ECML is a faster journey, albeit covering more miles. The journey planners give me routes with changes at Manchester (±Preston) or Lancaster via the WCML. S&C may be the fastest option in some cases when the line reopens.
Travel via Warrington doesn't seem to appear for Glasgow to Sheffield on the VTEC journey planner unless you specifically ask - it routes you either via the ECML or via Manchester. That said, times are broadly competitive with the more direct routes (although may well be overtaken).