How they work and why it doesn't matter
Not sure about MBenz in particular, but most such air pumps pump air (duh) into the exhaust right after the exhaust valve. The theory is that any unburnt hydrocarbons (especially if they were unburnt as a result of too rich a mixture) will burst into flame when exposed to an oxygen source since they are so hot (~1400F).
This is basically a burn-off of exhaust - outside the combustion chamber - so it adds no power when present and takes away none when it isn't operating.
The only downside I can think of is that if your mixture is too rich and the air pump doesn't burn off the unburnt hydrocarbons, they'll end up in your catalytic converter (if you have one/more) and that will shorten the life of the expen$ive cat(s).
Unless the belt that drives the air pump drives something else in addition, why not just remove the belt?