My question is how can they have the above technologies with only mid-1980s-level computers? Is this even remotely possible?
You are probably forgetting that we have been several times on the Moon in the 70's with computing power so small that it would make any today app developer laugh hysterically if they had to program with that in mind
On board Apollo 11 was a computer called the Apollo Guidance Computer (AGC). It had 2048 words of memory which could be used to store “temporary results” – data that is lost when there is no power. This type of memory is referred to as RAM (Random Access Memory). Each word comprised 16 binary digits (bits), with a bit being a zero or a one. This means that the Apollo computer had 32,768 bits of RAM memory.
In addition, it had 72KB of Read Only Memory (ROM), which is equivalent to 589,824 bits. This memory is programmed and cannot be changed once it is finalised.
To put that into more concrete terms, the latest phones typically have 4GB of RAM. That is 34,359,738,368 bits. This is more than one million (1,048,576 to be exact) times more memory than the Apollo computer had in RAM. The iPhone also has up to 512GB of ROM memory. That is 4,398,046,511,104 bits, which is more seven million times more than that of the guidance computer.
But memory isn’t the only thing that matters. The Apollo 11 computer had a processor – an electronic circuit that performs operations on external data sources – which ran at 0.043 MHz. The latest iPhone’s processor is estimated to run at about 2490 MHz. Apple do not advertise the processing speed, but others have calculated it. This means that the iPhone in your pocket has over 100,000 times the processing power of the computer that landed man on the moon 50 years ago.
If we have managed to send people on the Moon and get them back alive with that puny computing power, and we have also managed to build and run nuclear power plants and calculations for creating very sophisticated nuclear devices with nothing as performing as a computer of the mid 80's, it will be possible to find a way to control those technologies.
If I am not mistaken what you have with those technologies you listed is a problem of controlling, not computing.