I have a 2000 Silverado 1500 5.3 and I went through the same agony with the secondary air for over 18 months. Did all the exercises I was told to do, but all in vain. I finally tracked down a retired guru from GM who worked in the electronics side of the company and he informed me that from 1999 through 2002 the V8 engines had to run hot (190 degrees) in the first few minutes to close the loop on smog checks. The ECM had to see the down stream 02 sensors in a hot mode to tell it that it was burning off the carbon emissions. It was a crude system but it satisfied the AQMD people. He suggested that there could be three things that could affect the " task " completion and those were #1 THERMOSTAT....check and see if it's jammed open or a mechanic has replaced a high temp thermostat with a low temp. This situation caused more that 70 percent of the problems and a butt load of other bad readings...like EGR valves. #2...02 sensors should last about 200,000 miles. Test them with an OBD tester to see if they are reading OK. # 3....Catalytic converters go bad eventually and don't burn off the carbons, leading to the down stream 02 sensors giving low temp readings.
He told me to check the thermostat FIRST....and BINGO, some dumb A** mechanic had put in a 160 degree unit when it should have been 190. I bought a 190 and replaced the 160 unit and 3 minutes after starting the engine the secondary air completed and 15 minutes later the catalytic completed. Smog checked perfect the next day.