All of what you say is true, however it's not as bad as you might think. Once upon a time were these magical devices called "mainframes" [1], and in the 1960s they were quite powerful (for that era).
Most engineering work was done in FORTRAN, and it ran very efficiently on the hardware. There were (usually) no CPU cycle sucking GUIs to slow down the computers.
As a high school student in the early 1970s I was privileged to take a summer course at the Goddard Institute for Space Studies [2] where they had an IBM 360/95 mainframe [3] for the scientists to take turns using (job entry by punched cards, job output by paper printout).
It's been so many years, and it was only casually explained to me, but I think NASA used three other 360/95 mainframes (the IBM top of the line at the time) located at the Goddard Space Flight Center [4] in Greenbelt MD to track the Apollo missions. I think these ran the same program more-or-less in triplicate (but there was no hardware for syncing). I think NASA also had an IBM 7094 [5] running an independently written program as a backup in case something went wrong with the S/360 computers.
Trust me, these computers were very very capable for the time. It's not like the primitive computer onboard the Apollo LEM. Mainframes were quite up to the task.
[1] http://en.wikipedia.org/wiki/Mainframe_computer#History
[2] bonus points for whoever looks up that institution in Wikipedia and checks the picture to see what's located on the ground floor
[3] http://en.wikipedia.org/wiki/IBM_System/360#Table_of_System....
[4] http://en.wikipedia.org/wiki/Goddard_Space_Flight_Center
[5] http://en.wikipedia.org/wiki/IBM_7090#Notable_applications