Looking at the Neo4jTemplate class, its @PostConstruct method is the one that calls mappingInfrastructure.postConstruct, so in theory all constructors would react the same way. If your Neo4jTemplate is defined as a Spring Bean or spring-configured, then the BeanPostProcessor for @PostConstruct would call that Neo4jTemplate's @PostConstruct method (as long as the CommonAnnotationBeanPostProcessor is configured <context:annotation-config/>, which in turn calls the mappingInfrastructure's postConstruct method. If you are creating the template in code with new, then none of the constructors call the templates @PostConstruct method, and therefore none would call the MappingInfrastructure's postConstruct method.
So this might mean the issue is really with all the constructors. Even though something else is masking the other constructor's problem with this.