While it looks simple to the naked eye, combustion is a ridiculous chemical mess:
A detailed kinetic model of propane ignition and combustion in air is developed. The model includes 599 reactions with 92 species and invol ves both the high-temperatureand low-temperature mechanisms of oxidation.
This is expected as combustion of a hydrocarbon molecule does not occur instantly. When say, a propane molecule (C3H8) bumps into an O2 molecule, if they're not hot enough (ie, fast enough) to provide the activation energy required to initiate the reaction they will just bounce off each other and nothing happens. If the activation energy is available however, either from the molecules themselves or with the help of a bystander like a free radical or a catalyst, then the reaction starts. What happens depend on how the molecules collide, which means there are many possible combinations, and every step from "C3H8 + nO2" to "CO2 + H2O" is its own reaction with its own kinetics.
So you may ask "what is HO2 isn't is supposed to be H2O", well when molecules get whacked inside a flame with lots of free radicals in the vicinity they will rearrange in all sorts of ways, leading to intermediary products that are more or less stable (HO2 would be of the "less stable" variety since it's a free radical) which then react with whatever they can find, towards the final products of the reaction.
From the point of view of breathing humans, problems occur when the flame lacks enough oxygen to allow all the intermediary products to combust into CO2 and H2O. But this can also occur if the flame hits a cold object, like the bottom of a kitchen pot. This robs the gas molecules of their energy and prevents many reactions from occuring. The most unstable products will still react and end up decomposed into CO2 and H2O, but more stable molecules like CO will not react further with oxygen if the temperature is too cold.
I've dug up this old paper which contains lots of useful measurements.
Summary: burning gas in air generates a lot of carbon monoxide (CO) which then, under normal conditions, also reacts with oxygen and burns, producing CO2. However:
When the flame is yellow, combustion is incomplete due to lack of oxygen, which produces a lot of CO.
When the flame is blue but licks the bottom of the pot, it is cooled before combustion is complete, again producing high amounts of CO.
They find very little CO from an open blue flame in air, or when the bottom of the pot is far enough to not cool the flame before combustion has completed.
In a gas oven, you usually don't see the flame, so there is no way to visually check for this. Besides, the flame will be running against the sheet metal at the bottom of the oven, so it could be cooled by contact with the metal and produce lots of CO. In fact an oven combines both ways to produce CO: insufficient airflow due to dust plugging the vents, and flame being cooled on contact with metal. So I'd say chemistry agrees with CDC advice. There is definitely a risk of CO poisoning.
According to these papers, using a gas range for heating would probably not generate enough CO to cause problems, but it still is a very bad idea to use that as heating for many other reasons that have nothing to do with CO. If this is an emergency situation in the winter, people will probably plug up the ventilation and turn the room into an airtight container in an attempt to stay warm, which means excess CO2 or oxygen depletion are possible.
Also, as the second paper mentions, if you do a really good job of making the room airtight, when enough oxygen is consumed and concentration drops below about 15%, which is still breathable, combustion will stop and gas will be released into the room directly. That would displace the air and people would asphyxiate. Explosion would only occur when enough oxygen is brought back in, for example by opening the door while trying to get out. But it could also occur if, for example, the oven flame goes out first due to lack of oxygen, and there is no flame detector to turn off the gas, and the burners on the top of the stove are still burning due to having access to more oxygen.
Excess CO2 will also kill you, of course, but unless you're fast asleep you will feel it coming.
Besides, burning gas releases water vapour in the air, which will condense on the walls and ruin them, cause mold to grow, etc.
So, even if the advice is technically somewhat wrong about blaming only CO, it's still good advice. If fear of carbon monoxide prevents people from succumbing to good old asphyxiation and/or explosion, in a blackout situation where 911 isn't answering the calls, that's good enough.