I think it wasn't really hard to predict that computers will be small and available to literally everybody in about 30 years back then. At least not for people who knew about computers and their capabilities.
Note, though, that what Clark predicted here was that everybody would have a terminal to their "friendly local computer", which is actually one of the common mispredictions made in that era. The thought was that you'd have a few very expensive mainframes or minicomputers per city and that people would sign up for accounts on them and dial into them from a terminal at home.
One also has to be careful with quotations. The famous "640k memory" quote for example is famous but not a quote. Bill Gates always refuses to confirm that quote and there are also no real sources. I think that people like Gates were rather aware of how computers will evolve in future. He didn't need to adopt predictions from others.
Also, Gates, not being on the IBM Model 5150 design team, wouldn't really have been in a position to make any decisions about the 640k conventional memory limit. (And, in fact, conventional memory could go to 704 or 736k if only text mode graphics were used).
EDIT: Does anyone else find that Clarke's accent sounds surprisingly American (Given that he was, in fact, British)?