To continue your analogy: if the goal is to stop the overflow, then we should probably see what the contributors are.

If the faucet is on and your kid is dropping pennies in, both are contributing to the overflow but stopping your kid is going to be irrelevant.

Do we know which humans are?

The point is that the faucet, in this analogy, is not contributing to the overflow. The natural inflow from the faucet is matched by a natural outflow from the drain. The surplus is the problem, not the absolute inflow.

Of course, turning down the faucet would still help. But in this analogy, the faucet is jammed on, but the kid might stop if you ask him the right way.

So your assertion is that the planet is a perfectly steady state, static system?

Almost. Right now, the drain is actually emptying the tub faster than the faucet is filling it. Atmospheric CO2 is increasing at a lower rate than CO2 is being released by humans, because natural sinks are absorbing a lot of it. It was fairly stable, on human timescales, before we started adding our own input.

Can you substantiate your thesis a bit? I.e. explain where the CO2 that the nature is (allegedly) producing is coming from?

For humans, it's pretty simple: we're burning the oil on the timescale of centuries that the nature has been storing on the timescale of billions of years. For nature, the longest CO2 cycle is at most a multiple of the longest life-span (~500 years), and there's no indication that the cycle has changed (i.e. that suddenly more trees are burning in natural fires than have for the past few millenia).

According to the EPA:

"While CO2 emissions come from a variety of natural sources, human-related emissions are responsible for the increase that has occurred in the atmosphere since the industrial revolution."

Ref: https://www.epa.gov/ghgemissions/overview-greenhouse-gases (on the carbon dioxide tab)

And the Energy Information Association goes on to include magnitudes here:

http://www.eia.gov/oiaf/1605/archive/gg04rpt/pdf/tbl3.pdf

If those numbers are accurate, humans contribute 0.2% of CO2 and 60% of methane. So my follow up is: Which gas is a worse contributor? Because we can greatly influence one of those.

But "source" of CO2 is meaningless without the other column, "absorbtion". Obviously nature produces CO2 (e.g. by animals and plants breathing), and CO2 isn't a problem by itself. The problem is the increase of CO2, i.e. the delta between production and absorbtion. That 0.2% of human CO2 surplus turns into 20% increase over a 100 years. That's the problem, not the production/absorbtion/cycling of CO2 itself. And that part, AFAIK, is all human-made!