No, my daughter's school does NOT allow this... for the reason you stated.
My daughter's school is a public school.
At my daughter's school... they do NOT even allow, parents to "request" a Teacher. Period.
AND... the classes are RANDOMLY arranged. ONLY the Principal... does the classes. Randomly. AND he will only consult with the Grade Level Teachers... about children that are a "concern" or may need to be put with a certain Teacher... for their benefit. But it is NOT based on race/gender/socioeconomic or academic levels.
At my Daughter's school, Parents are NOT given the power or the choice, to "choose" the Teacher or class their child is in. Because as you see... it causes BIG problems.
AND it benefits the children, by being in different classes than their "buddies." Because otherwise, they do not learn how to get along with others, they form negative cliques (and the Moms too) and it is just a big mess of cliques. NOT good.
My daughter's school does it this way because... ONCE the children are in middle school... they would have met/got acquainted with/mingled with MANY of the kids in their grade level... and THUS, once they go off to middle school... they then while there... can know and HELP each other.... and look out for each other.
WHICH to me, is a GOOD thing and approach... to how my daughter's school... arranges the classes.
My daughter, while in 1st grade, did not have one single "friend" in her class... she is on the shy side... but so what. She did NOT mind, she did fine, she made friends and was perfectly adjusted and liked her class and Teacher. She did well and was confident in class....despite having all new faces. She now knows... more kids, than before.
all the best,
Susan