Citation? The original article advocates reform, and possibly abolishing permanent secretaries, but nothing like entirely abolishing the civil service
https://dominiccummings.com/2014/10/30/the-hollow-men-ii-som...
http://theconversation.com/dominic-cummings-wants-to-add-mor...
https://www.theguardian.com/commentisfree/2019/aug/01/boris-...
The institution will obviously continue to exist under its current name.
1) continuity of government function in presence of political instability, elections etc.
2) higher level of comptency across government employees ( career bureaucrats)
Essentially you don't want the government to be staffed with incompetent friends and affiliates who either leave or get kicked out after lost election leading to the vacancies, interrupted projects etc.
More: https://en.wikipedia.org/wiki/Civil_service#Civil_service_in...