I find the East - West link curious. The National government continually opposes rail on the basis of the economic return. The East - West link has a business case analysis (BCA) of 1.9 at absolute best case (more likely around 1.3 depending on what numbers you believe). It normally takes a BCA of 3 or greater to get a roading project up off the ground. I don't oppose the project but it appears in this case that there is more at play here than other projects.
The Road Transport Lobby (Trucking Companies) are very strong and influential.
As for the GA Rapid Intercity Rail idea - I like it. I don't think they should bother with Stage 1 though - I fear it would put people off. Go straight to stage two.
It requires a third and fourth main through to Westfield (and maybe further), but they are going to be needed anyway. If the trips are priced right, their end projections of 2 hours to Tauranga and 70 minutes to Hamilton are great (and would be easier than taking a plane). It would be used by tourists as well as commuters.
I do think their costings are way too low, but instead of spending $10bn on 10 more 'Roads of National Significance' how about sharing some of it with rail (Wellington could also do with Rapid rail to Palmerston North).