This article has a bunch of cringeworthy prose, but has some worthwhile graphs of data showing that price-fixing isn't the answer:
https://www.bloomberg.com/opinion/articles/2021-03-02/berlin...
See also numerous Planet Money stories about rent control:
https://www.npr.org/sections/money/2019/03/05/700432258/the-...
https://www.npr.org/2022/01/31/1077086398/is-it-time-to-cont...
https://www.npr.org/2019/03/29/707908952/the-evidence-agains...
Note that even very-progressive Jerusalem Demsas was once against rent control and has only switched sides as a palliative measure because fixing the root cause of the problem is proving too difficult.
Freakonomics also did a show on the topic:
https://freakonomics.com/podcast/why-rent-control-doesnt-wor...
https://www.lewis.ucla.edu/programs/housing/housing-supply/z...
edit: So I actually consider rent control to be a result of low housing production rather than the other way around. Associate Professor Shane Phillips from the Lewis Center (linked above) also considers rent control to be a very reasonable policy as long as there's plenty of realistic zoned capacity for more housing. Everyone is always arguing that if we build more then that would keep rents from rising so why not have rent control anyway then?
But yes, I think you're right that rent control is typically a political result of insufficient housing production. But it's a bad policy. Price fixing is never the answer to a supply problem, even if it's a popular one.