The Accountable Care Act (Obamacare) specifically mandated that routine preventative care be covered by private insurance plans at no cost to the patient. That way patients won't skip preventative care to "save" money and problems can be detected early before they become serious.

https://www.kff.org/health-reform/fact-sheet/preventive-serv...

What you propose would work fine if humans were fully rational and not living paycheck to paycheck. Unfortunately that's not reality.