The reason that econ-101 type reasoning does not give the right answer is because the conditions for it applying do not obtain. This is a fact that has been well known since at least Hayek's Road to Serfdom.
Governments like price controls because they regard their success or failure on a very different basis to what ordinary people do. Governments do not particularly care about shortages. Governments care about what they believe will keep them in power, what will give them more power, what will extend their control over more parts of society. Thus the primary feature of price controls that a government will be concerned about, the metric by which their success will be judged, is compliance.
Prices are signals of information. They indicate the degree to which people desire things relative to other things. Governments desire things that are quite distinct to what run-of-the-mill people desire.
People desire such things as food, shelter (including heat, light, water, etc.), clothing, medicine, transport, etc. These are things that keep people alive and healthy. People who are not so motivated do not live long.
Governments primarily desire to stay in power. Governments that are not so motivated do not stay around very long.
Thus, price caps may meet the desires of the current government, while cruelly disadvantaging ordinary people. Indeed, it is the usual expectable situation.
Price caps on fuel, for example, can be held up as "doing something to help the people." They can also be used as leverage on various individuals in order to wield additional power. Oil companies have huge wealth and so potentially have political influence. A chance to manipulate them with the threat of price caps gives the government additional levers of power. Nice oil company you've got there. Shame if the price of oil were half. But do us this favor and we'll let you keep 90% of the current price. For now.
The favors from the oil companies can be traded with other sociological entities for favors from them. The transit worker's union can be manipulated by agreeing to make driving private cars more expensive, thus pushing more people onto the subways. The agricultural sector can be manipulated by lowering the price of fertilizer (which is heavily affected by the price of fuel). Or at least not raising it as much as anticipated. The building and trades unions can be manipulated with promises of caps on the price of houses (affected by the price of fuel) to supposedly stimulate the house building market. And so on.
Thus we see price controls implemented again and again. And when government A is tossed out to be replaced by government Not-A, why the new rascals ask if they can borrow the notes from the previous rascals. Because the new rascals want to implement the policies that only last week they were shouting about being the most evil and ill considered nonsense.
This is by no means new. For example, in the late 1970s it was the case in Canada. The Liberal govt brought in price controls to "fight inflation." Then they lost an election and the Conservatives were elected. Who immediately implemented virtually identical price controls. Which managed to make things worse.
And so the price controls on fuel require price controls on housing and transport. Which require controls on wages in trades. Which require controls on the next aspect of society and the next and the next.
Hayek told us all of this quite clearly almost 80 years ago.
At some point, one expects that the following motto will be heard: Real price caps have never been tried.