While Algebra 1 in 6th grade may be typical at schools for the gifted, plenty of kids apply and are admitted to prep schools from ordinary middle schools as well. These kids take 6th grade math curriculum and then pre-Algebra in 7th or even eighth grade. While I bet that most students applying to selective boarding schools have had some Algebra and many have had some geometry, I also bet that all of those schools offer at least one flavor of an Algebra 1 course for kids who have not had any real exposure to algebra before starting high school along with offerings for students who have had algebra but need a review. This will be particularly true for another year or two while the effects of the covid lockdown are still rippling through students who lost significant learning time.
All of that said, I think the OP student should try retaking the SSAT after some studying or even tutoring. I think that if the student is able to get the quant score up to 65+, their application will be in stronger shape. An overall 87th percentile is absolutely fine in general, but depending on the school, having a 50 quant score is probably low enough to raise some concerns that the student may struggle with the math curriculum.