Working with your example: If we assume the government health service is behaving in its own self interest, why would it spend money on test that it knew not to be cost effective? Whereas if the incentives are split between a Dr ordering a test and an insurance company paying for one wouldn't they disproportionately order tests?

More generally, even if its true for particular testing procedures theres lots of low hanging fruit for intervention before things get severe. The most obvious examples would be schemes to get people to stop smoking or lose weight, that the government provides freely because they are less expensive than the projected cost of the illnesses that would arise without such intervention. Also the ability to see a general practitioner more regularly than if you paid per visit means symptoms can be picked up earlier (e.g. if someone has a mild symptom but has to pay to get it checked they are disincentivised to get it checked until it becomes severe.)