I also thought this was the case, but everything I've seen suggests that taking melatonin does not alter the natural production of melatonin.
You are correct about everything else though.
Similar inefficacies have been seen clinically e.g. in [3] and are (caution, anecdata) widespread on the internet, with "melatonin doesn't work" being a popular search term with tons of articles about it. An honest to goodness test seems to have been done at [4] where they made sleep disturbance symptoms "disappear" by resuming treatment at a lower dosage, but instead of blaming the neurons they are blaming the liver, saying that it got overloaded and couldn't clear melatonin out of the bloodstream anymore in some patients—I just want to include that as a plausible alternative explanation so that you don't take my words as gospel truth or anything. I’m trained as a physicist, not a physician, and there is this meme of people with physics degrees thinking that everybody else’s field is their expertise and like I want to be deliberately self conscious about my limitations here.
[1]: PDF Warning: https://www.herbogeminis.com/revista/IMG/pdf/melatonin-adhd....
[2]: https://academic.oup.com/sleep/article-abstract/33/12/1605/2...
[3]: https://www.psychiatrist.com/jcp/effects-exogenous-melatonin...