Reading that information leads me to believe and agree that there is much more to be learned about the behaviour and the climate of this planet.

Several facts that are presented in the article really got my attention.
One was this one:

A few years after Arrhenius published his hypothesis, another scientist in Sweden, Knut Ångström, asked an assistant to measure the passage of infrared radiation through a tube filled with carbon dioxide. The assistant ("Herr J. Koch," otherwise unrecorded in history) put in rather less of the gas in total than would be found in a column of air reaching to the top of the atmosphere. The assistant reported that the amount of radiation that got through the tube scarcely changed when he cut the quantity of gas back by a third. Apparently it took only a trace of the gas to "saturate" the absorption — that is, in the bands of the spectrum where CO2 blocked radiation, it did it so thoroughly that more gas could make little difference.

Another one was this:

Still more persuasive was the fact that water vapor, which is far more abundant in the air than carbon dioxide, also intercepts infrared radiation. In the crude spectrographs of the time, the smeared-out bands of the two gases entirely overlapped one another. More CO2 could not affect radiation in bands of the spectrum that water vapor, as well as CO2 itself, were already blocking entirely.(8) These measurements and arguments had fatal flaws. Herr Koch had reported to Ångström that the absorption had not been reduced by more than 0.4% when he lowered the pressure, but a modern calculation shows that the absorption would have decreased about 1%

Here the author seems to put more emphasis on a modern calculation that a older measurement.
Would it not be more scientific to make a modern measurement instead?

The effect of the temperature on the CO2 and climate are interesting.

The rise and fall of temperature was tied up in a complex way with interlocking global cycles involving not just the mineral geochemistry of CO2 in air and sea water, but also methane emissions, the growth and decay of forests and bogs, changes of the plankton population in the oceans, and still more features of the planet's biosphere.

This information is also interesting.

The CO2 levels in their record got as low as 180 parts per million in the cold periods and reached 280 in the warm periods, never higher. But in the air above the ice, the level of the gas had reached 350 — far above anything seen in this geological era and still climbing.

Computations are good, but is this computation based on a comprehensive input information?

The computations pinned down an imbalance. The Earth was now taking in from sunlight nearly a watt per square meter more than it was radiating back into space, averaged over the planet’s entire surface. That was enough energy to cause truly serious effects if it continued.

I'm wondering about your posted hypothesis:

CO2 in the atmosphere acts like the glass in a greenhouse. It reflects back re-radiated energy from the surface of the earth, rather than letting it pass back out into space. Expanding gas doesn't really enter into the picture.

Gas is very interesting stuff, it usually gets colder when expanded.
See HERE and
HERE.

If CO2 does reflect infrared energy that comes from the planet back to the planet , then why would it not reflect the same amount of infrared energy that comes from the sun back into space?

I still think that my hypothesis stands a better chance of explaining all these global changes.