Puberty is just the physical changes — bone growth, the growth spurt, secondary sex characteristics, brain maturation — that estrogen or testosterone cause in a developing body. It doesn't matter where the hormone comes from.
I'm going to borrow your words to me: this is simply wrong on the biology and medicine. Specifically, the words between your hyphens/em dashes are correct and the words either side are not. It's been explained already by a PP that puberty requires more than just oestrogen or testosterone.
Your claim that it's "impossible to go through the puberty of the opposite sex" is doing political work, not describing biology.
No, it's describing biology. Someone with CAIS will not go through a normal female puberty when given oestrogen. For example, as you say, there is no uterus and the person in question has testicles not ovaries. There is no normal female puberty in the world that involves testicles. The important thing here is that there has been no disruption to the hypothalamus releasing the Gonadotrophin Releasing Hormone (GNRH), so the normal pubertal development pathway is still firing, despite the CAIS DSD.
No one claims these women are stuck in a permanent pre-pubescent state, failed to mature into adulthood, or suffered brain damage from having their puberty driven by external rather than gonadal hormones.
Puberty has been driven by GNRH, just like it is for everyone else because nothing has blocked it from getting started. Considering no testosterone can ever be processed effectively, and the person with CAIS most likely assumed themself to be a normal female until there was no observable sign of pubertal development, the only hormone that makes sense for bone health and secondary sex characteristic development is oestrogen. Because the process of puberty wasn't blocked from starting, there is no indication that there would be brain damage.
Also DSDs have absolutely nothing to do with gender identity.