I think you're overestimating the impact of the Morris Worm - there weren't all that many servers to bring down.
The first commercial ISP in the U.S. was founded in 1989.[1] Until then, only a few people in universities and the tech and defense industries had access to the internet. According to this chart[2], the number of internet hosts in 1992 is barely distinguishable from zero on today's scale.
[1] https://en.wikipedia.org/wiki/Internet_service_provider#Hist...
[2] https://en.wikipedia.org/wiki/File:Number_of_internet_hosts....
In any case, there was definitely a network that was being used by a lot of people by 1988.