The tailpipe test checks the tailpipe, not the oil pan. Does the emissions test facility stick their probe down the dipstick tube?
That's the part I struggle with, what connection there is between emissions testing of the exhaust and how that's influenced by the oil. Even "bad" oil.
You heard of the PCV system? Where crankcase vapors get recirculated back into the intake to get burned in the engine. Since the engine isn't 100% efficient, some of that is unburned and goes out the tailpipe. The more you start with, from old oil, the more will get sniffed in the tailpipe test.