1. The Western Alignment would not be cheaper, going from Manchester to Glasgow by the straightest possible alignment via Carlisle is ~305km, with a spur to serve Edinburgh being ~70km if you really want a reasonably direct route above all else.
That means 375km of track.
The Eastern alignment would go via Newcastle, I will include York and connecting to Glasgow via Edinburgh (with an inland alignment from Newcastle to Edinburgh), this comes out to a similar count of 370km or so of track
Additionally the terrain for a significant fraction of the Eastern alignment is nice flat effectively tableland with only a hundred miles of "difficult" terrain compared to 200+ for the western alignment (crossing the Lake District
and the borders rather than just the latter).
2. While it is true that trains to Glasgow would be faster via Manchester than Leeds, the difference in track is only roughly 60km or so between the two routes.
This means that a train doing 320kph (as the train would simply cruise for longer at its top speed) would eat up the extra distance in
eleven minutes. This is hardly significant on the end of a ~3hr journey.
3. There is insufficient traffic on the western alignment. If you look at a
map of the UK's population density you can see that between Manchester and the Scottish Central belt is nothing, while the eastern alignment has Newcastle and its surrounds.
A Scottish HS2 link would never be viable on just traffic to points south of Manchester, two 400m trains per hour splitting at Carstairs or similar would easily saturate demand even if the air market was completely obliterated which is unlikely as many Anglo-Scottish flights are connections to international ones and as such to not suffer from security check delays.
You can boost demand significantly by shifting the Newcastle traffic, which is similar if not larger than the Scottish traffic due to the rail network already holding almost all of the modal share. Travel times under 2hrs to London are a possibility.
4. You would either have to abandon Newcastle to become even more of a backwater or you would have to build the line as far as Newcastle anyway, resulting in the Scottish link becoming a complete boondoggle.
HS2 Ltd. wants the western alignment so it can justify a Newcastle link later on, additionally it has unlimited money in said current climate (where politicians will spend ludicrous sums on "green" transport policies).
The Eastern alignment also gets you a fourth connection between Glasgow and Edinburgh that would make a shuttle service with 19 minute travel times feasible (based on existing Shinkansen timetables on a line with a 270kph top speed that requires tilting trains to exceed 250)
Why do I have to keep making this argument?