This is the error message that I got (not engine lite):

P0410: When the engine is just started, secondary air injection system mandatorily import air into exhaust manifold to oxidize toxic gas generated such as hydrocarbon. The process also accelerates the warm-up of catalytic converter.

So obviously my car didn't shut down because of the air pump. This is what he wrote in the quote:

-New Air Pump and Relay: $1,250
-Remove and replace air injection reactor: $38
-Relay: $360

They didn't check RCM because the mechanic was not qualified for that...