For heating system, you're assuming that the burner unit is regularly maintained (i.e. cleaned of soot) which is a pretty far stretch - most people don't do anything until the unit breaks down, usually because the flame-out protection sensor is so covered in soot that it can't detect the presence of the flame any more.
For stoves, gas stoves are being banned for new construction because they don't burn as cleanly as once thought [1].
For BBQs the heat source used doesn't matter (with the exception that coal units should never ever be used indoors), they all emit tons of toxic and highly odorious particles from all the oil and fats decomposing.
Wood and charcoal, yes. But natural gas and propane are clean burning. The difference between them are several orders of magnitude.