As far as i remember the station has always been pretty grim. Half painted walkway over the road (can't blame Infratil for that, but a few cans of paint would have sorted it), puddles inside, lack of shelter for waiting passengers etc, cheap nylon carpets etc. It could be a great asset for PIK, but the lack of even basic maintenance is appalling, if they want to attract more people to use rail to the airport (as is intimated in the draft master plan) then surely getting it up to speed would help.
The station is on of the few in the UK not owned or operated by National Rail or a Train Operating Company (in this case it would be Scotrail), so PIK and Infratil are responsible for it's maintenance. I'm assuming that includes the walkway.
As for your point about EDI. I'm sure it's going to attract a lot of interest, the busiest airport in Scotland, and with Edinburgh being a major European city for business, education and tourism, whoever owns the gateway to those markets could have something very valuable indeed!