The problem with the fire department is the freerider problem - if I pay for the fire department’s “protection” but my neighbor doesn’t, and his/her house is on fire, the fire department has to save my neighbor’s house in order to save mine, so the neighbor is not incentivised to pay.

Healthcare doesn’t have this issue (except in emergency services), so public healthcare also doesn’t solve that. It solves problems such as the principal-agent problem, asymmetric information, improves collective bargaining, ...