The flushing in menopausal women isn’t so much caused by the low estrogen as it is the body’s and the brain’s (hypothalamus) response to dropping estrogen levels. This is reviewed here but is a complex business.
When estrogen levels drop in a post-pubertal woman, her hypothalamus tries to re-stimulate estrogen production by secreting various proteins (peptides). It is the release of these and other neuropeptides, either directly or indirectly, that leads to flushing since in addition to stimulating estrogen release, they also affect the sensitivity or ‘gain’ in the body’s temperature regulation system, and that makes for unpredictable episodes where the blood vessels dilate to give off heat.
It helps to remember that many ‘vegetative’ processes in the body, such as body temperature regulation, appetite, puberty, thirst, emotion, etc. are controlled by the hypothalamus, so spillover effects may not be completely unexpected when there are surges in neuropeptide production.
Your question is a very interesting one but the real mystery is what controls (the onset of) puberty and how passing through puberty resets the hypothalamus.