Linux can be as small as you want. Look at Damn Small Linux, for example. You can actually build a kernel to run on certain CPUs with no memory- management hardware.
There was actually a whole lot of computing power on the ground, in the form of those IBM mainframes, backing up the guys in space. Everything they did had to be managed in coordination with ground control, all the steps worked out in advance.
Bit more Government management of the process back then, bit more trust to private corporations like Boeing, with its stellar reputation for relentlessly pursuing reliability over profits, these days.
Coincidence? You be the judge.