Not only is this statement true, but the amount of solar (sun's) irradiance or heat leaving the sun does vary. A well followed example is the number of sunspots present on the surface of the sun at any one time and the length of the sunspot cycles. More sunspots, which have bright, hot haloes around them, correlate well with a warmer surface of the Earth. Fewer sunspots generally correlate with cooler times. Also, the length of the sunspot cycles correlates with Earth's temperature (see figure below). Trying to forecast the climate changes with models that do not include solar variations cannot be expected to result in reliable outcomes or ranges of outcomes.
That no one factor correlates perfectly with whether Earth's surface is warming or cooling is expected due to the fact that, while the sun is putting out a little more or less heat, Earth's distance from the sun in its orbit around the sun may be changing or the amount of cloud cover on Earth may be changing and exerting an opposite effect on the climate. Some of the drivers are predictable, like Earth's orbital variations, while others, like solar intensity or volcanic explosion are not predictable at this time. Regarding atmospheric CO2, as oceans become warmer they release more CO2 into the atmosphere, as does a bottle of carbonated water as it warms. As oceans become cold they tend to absorb more CO2 from the atmosphere.
In summary, the sun is Earth's ultimate heat engine and thus plays a dominant role in determining global warming or cooling. Changes in ocean currents also affect earth's climate in a major way but oceans receive their heating predominantly from the sun.