Not true at High schools where BC always has to follow AB. BC purposely starts where AB left off. These high schools do not allow BC after Precalc because the BC course does not cover AB.
I do agree with you that summer review on their own is a good idea before the school year starts.
OP: your kid’s C in first semester will not matter at all, given that they ended with an A. Even if colleges can see the C , they will see the amazing upward trend! It shows resilience and a growth mindset. If they love stem, press on with stem!