Both the F-104 and Mig-25 are on the record as being very high-performance, record-setting zoom climb machines. There are some other planes out there that can do a similar job. I believe the F-4, oddly enough, set some records in that area using a slightly modified engine setup (water injection).
The F-15 would sort of be the textbook platform for this: there's a reason they used it to launch ASAT missiles, and you could fly one today with more powerful engines than that eighties-era F-15. On the other hand, part of Starfighter's appeal is that they can do these launches relatively cheaply.
> Mach 3.2 vs 2.0 for the F104
You're comparing a pretty desperate MiG-25 speed to the pursuit speed in the F-104's flight manual. Speed in level flight isn't really the issue here anyhow.
EDIT: Starfighters indicates their Italian F-104s will do Mach 2.2 while climbing: http://www.starfighters.net/fleet/
Lots of good stuff on that page.
In contrast J79 engines and parts are available on ebay and knowledge of how to maintain them us fairly common, just check Youtube!
They may be using F-104s because there are several in private hands (Wikipedia claims that the FAA says that there are 10 in the US) with owners willing to do satellite work.
The Air Force and NASA used the planes for high-altitude test missions, and so there is some comfort in outsourcing those missions to the company. I've always been under the impression that most of their business was government stuff, but I could be all wrong about that.
Even the MiG 29 would get you marginally better Mach numbers (2.2 vs 2.0) and better payload. Plus they were produced in similar numbers as the F104.