Assuming that there had already been a moving brake-test on departure from Blackpool, why was another one necessary from Preston?
Because there would likely have been a crew change at Preston, meaning that the new driver had to establish the performance of the (unfamiliar to them) unit using the running brake test (RBT).
If two brake-tests are deemed to be imperative, why was one not required leaving Wigan?
Because Wigan isn't a crew change location for Avanti.
Timetable-compilers must be aware of the brake-test requirement, so why don't they make provision in the schedules?
You raise a good point. For Avanti, LNER, XC etc. services, where there will almost always be a crew change at certain locations (e.g. Preston, Newcastle, Leeds), IMHO it would make absolute sense to mandate an adjustment allowance after departure, to account for the inevitable RBT. Indeed this is already mandated in the Western & Wales region for GWR services on 80x timing loads.
I suppose the primary logic for not having such a rule elsewhere is that the timings and headways should have sufficient built-in 'slack' to recover such micro-delays - and this change was, AFAIK, brought about in connection with the lower headway (2 mins) which was adopted on the line from Paddington to Reading, in preparation for the GWR December 19 timetable change. This left less room between the planning headway and the technical headway. This is in contrast with the WCML for example, which has a technical headway as low as 90 seconds, even on some of the 125mph stretches, but is still planned on a minimum of 3 minutes.
Moreover, in many cases, introducing adjustment allowances would not be possible without rewriting the timetable. So even if the rules changed, they would probably have to 'grandfather in' existing services that wouldn't work any other way.
It's also worth noting that some services will have many potential crew relief points - e.g. a Manchester Airport to Barrow service could have crew changes at any of Manchester Oxford Road, Wigan, Preston or Lancaster, and the most efficient relief point will vary by train. Therefore, Operators don't want to tie themselves into relieving at one specific station when they are bidding the timetable, as they will not start producing crew diagrams until they have confirmed timings for all the trains in the timetable. And equally, it is very unlikely that all possible relief points would be utilised on a given train.
Notwithstanding all of the above, there are proposals to introduce driver change adjustment allowances in some areas, and I'd say it's not beyond the realms of possibility that it could end up extending beyond the Western region.
Why haven't Avanti's performance managers not insisted on 1-minute later advertised departures from Wigan and Warrington?
To maximise punctuality, you want advertised departures to be as early as possible, and advertised arrivals to be as late as possible, so that you can "underpromise and overdeliver" (or, more realistically, to allow the train to leave a little bit earlier if it's ready, whilst leaving some "slack" in case of delays).
Of course that must be balanced with commercial considerations in terms of journey time. But - passengers are really just interested in whether they arrive on time, not whether they depart late and make up some time.
Arrival at Crewe is scheduled at 13:00, but the advertised arrival time is 12:57. So if the train were actually on-time it would appear to be 3 minutes late! Why is this?
There are a number of Avanti and other operators' services which are advertised to arrive before they're scheduled to do so. In this case, I can see no obvious reason for it, and the train is virtually guaranteed to arrive late as you say. However, in other cases there can be good reasons for doing so.
For example, journey planners etc. do not like services that are advertised to depart before they arrive - they may throw up errors. So if there is a standard 15 minute allowance for Single Line Working on a given part of a route (say Newcastle to Durham), you wouldn't want trains to sit around for 15 minutes at Durham on days where the Single Line Working doesn't occur, or is further south.
So you might decide to advertise the arrival and departure at Durham 15 mins prior to their scheduled values, to allow for an immediate departure on days where the SLW doesn't materialise, and accept the cost of Delay Repay on the (rare) occasions where it does happen. You would then go back to advertising the booked times at the next major station (e.g. York), or at the end of the SLW.
Incidentally, we had yet another moving brake-test after departure from Crewe!
It's possible there was another crew change, although as far as I'm aware, most of the Blackpool/Edinburgh via Birmingham services only have crew changes at Preston and Wolverhampton or Birmingham. Drivers are authorised - encouraged, indeed - to undertake RBTs if they are ever in doubt of a unit's braking capabilities, for example if they suspect low adhesion due to the onset of rain after a prolonged dry spell.
It's also possible it was simply a TSR or ESR, although I'm not aware of any in place immediately south of Crewe.