The chemical reactions in the exhaust system are caused by specific byproducts interacting with the catalyst embedded in the converter. A catalyst is a compound or a component which causes or assists in a reaction without being consumed.
There is no connection between fuel economy or engine lifespan and the converter, other than that the engineers responsible for designing and building the vehicle, specifically the engine, can take into allowance that some of the chemicals generated by combustion will be processed/eliminated in the converter.
As a result, it is likely that less effort is expended in an attempt to prevent these harmful compounds from entering the atmosphere. It is also likely that the results will allow the engineer to focus on fuel economy and engine longevity. The linked site goes into much detail and also notes that a failed converter will create back pressure which is going to adversely affect the engine performance. Image from a UK site: