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1. It's not an "air pump". It's an A. I. R. pump,
Air
Injection
Reaction. GM does love it's acronyms, although I generally leave out the periods and spaces. The name gives a clue to it's purpose--
supporting a chemical reaction by adding air to the exhaust stream.
2. "Diluting" the exhaust with ordinary air is not it's purpose.
3. AIR pumps were on vehicles years before catalytic converters. The intention of the thing depends on the overall emissions-reduction strategy.
The purpose of AIR on my '88 K1500 is to warm-up the exhaust system, in particular the catalyst, immediately after engine start. Catalytic converters don't work when they're cold. The oxygen sensor doesn't work when it's cold. The rich mixture provided by the cold-enrichment system means the exhaust gas is not fully-reduced. In other words, the exhaust gas is capable of supporting combustion even after it's being blown down the exhaust port--all it needs is additional oxygen squirted-into the exhaust manifold to turn carbon monoxide (CO) to carbon dioxide (CO2); and if there's still hydrocarbons (HC) in the exhaust, they'll reduce to water vapor (H2O) and CO2. In the process, that combustion produces additional heat that warms everything down-stream.
The AIR system shouldn't pump air into the manifold once the computer is ready for closed-loop operation--the additional oxygen from the air being pumped into the exhaust manifold will screw-up the O2 sensor readings. So the output of the AIR pump is diverted into the air cleaner. But by then, the exhaust system is warm enough that the catalyst is functional.
Other uses of the AIR pump involve "full time" reduction of combustible exhaust, exactly as described above. In vehicles that don't have O2 sensors, (My high-school '69 Chevelle, for example) the air was blown into the exhaust manifold almost the whole time the engine was running. There was a vacuum-sensing diverter valve that stopped air flow into the exhaust system during deceleration(high manifold vacuum.) Air in the exhaust during deceleration leads to exhaust backfires that blow the muffler apart. I had two blown mufflers on my Chevelle for this exact reason.
A third use of the AIR pump/diverter valve involves "upstream/downstream" switching. "Upstream" air warms the exhaust system as before, getting the catalyst and O2 sensor (if used) toasty, quickly. Then the system would switch to "downstream" injection, into the catalyst itself to provide oxygen for the reaction within the cat converter. Downstream injection was always far behind the O2 sensor, (if used) so it didn't screw up the signal. Typically, the air was blown into the catalyst behind the first section but ahead of the second section. The first section reduced hydrocarbons and CO, the second section worked on oxides of nitrogen (NOx)
The AIR pump (in good condition) doesn't use enough horsepower to make a difference. You'll never see it on the fuel economy without real instrumentation. They can go bad, bearings seized, etc. so they become hard-to-turn, noisy, etc.
The diverter valves tend to cause more problems--sticking in one position instead of switching back-and-forth like they're supposed to. Exhaust backfires result from sticking one way, if it sticks the other way on an '88, you'll just have slower exhaust warm-up.
In general, the system is simple and reliable. No reason to disable it, or to be scared of it. If it's broken, it's easy to fix.