I would argue that such the administrative state has been in existence since long before the computer. Its existence does not rely on any particular technology other than the ability to write things down (in whatever way) and to read them. The administrative state did not really exist at the time of the founding of our country, but has gradually grown as the government has come to be more involved in our lives. It began to grow in a very serious way about 100 years ago.
The existence of the administrative state goes along with political facts, not technological ones. When a country wants its government to be more deeply involved in its society and economy, the administrative state grows. The existence of computers has helped the administrative state to grow, but that state was definitely in existence before computers existed.
The administrative state really started to grow during the Progressive Era when there was a push for more government regulation of the economy. It grew more rapidly during the New Deal when the government started to try to do more to direct the economy. It grew even more in the post-WWII period as Americans came to accept the idea that the government would be involved in more parts of their lives.
Thus, the administrative state existed before computers. It started to really grow about 100 years ago and has become more entrenched over time.