Nothing pays for Comprehensive Sickness Insurance. EU Directive 2004/38/EC
is very specific about when Comprehensive Sickness Insurance is required (exercising treaty rights as a student or self-sufficient person) and when it is not required (working either as employed or self-employed or seeking work).
Remember that nothing is paying for your Comprehensive Sickness Insurance when you are seeking work. The seeking work period must not exceed six months.
And remember that if the EEA citizen is not exercising the treaty rights listed in the Directive, technically they have no legal right to reside in the UK. That right is contingent on their exercise of treaty rights.
It could get funny if the Home Office starts declining EEA citizenship applications on the grounds of the EEA citizen being illegally in the UK due to not exercising treaty rights at any time in the past ten years.
As an aside, the DWP only considers you as working for the purposes of exercising treaty rights if you earn more than £153 a week, the threshold at which you start paying National Insurance. The Home Office has not adopted this interpretation and considers you working even if you are not paying National Insurance, so long as you have an NI number and are registered with HMRC as employed.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.