Public services are not projects that can be signed off. They are ongoing, maintained services whose users needs will change over time. These needs should be at the core of a service and not an afterthought. Money and resources to be provided throughout the lifecycle and not simply provided upfront.
Combining agile and non-agile approaches creates confusion and ultimately leads to failure because agile does not require any other approach to be used with it. Agile is not a set rules that must be followed, it is a mindset that allows for the requirements of a project to change over time, in accordance with the needs of the end users. This is especially important with public services such as healthcare, as it is natural for both government expectations and user expectations to change throughout the life of a project. Agile not only allows for this to happen but it also gives people more control over what is happening. To try and combine this with an approach such as the waterfall-type methodology leads to failed outcomes because the waterfall targets and users needs are unlikely to be met. Using agile on its own is therefore the most effective way of implementing change in public services.