Depends what you mean by “allow” – the CIFSS rules say that there can’t be “undue influence” from sports. Given the number of Top 10 in the nation football programs (Mater Dei, St. John Bosco), basketball programs (Sierra Canyon), and baseball programs (Orange Lutheran, Harvard-Westlake) in SoCal, I can assure you that some amount of influence is apparently due, lol
Oh, no doubt. I live near De La Salle high school in Nor Cal - historically a national football powerhouse. Some schools definitely seem to do well despite “not recruiting.” 
Still, for small, academic schools, especially but not exclusively boarding schools, they need kids who check a lot of boxes. A sports phenom who is all about one sport is less likely to find the teams up to their standards, and they won’t be able to fill out the school’s orchestra or debate team - not a good match for either the school or the athlete.
Cate for example has 150 boys total, and each of them has to play a sport each trimester. There aren’t enough pitchers to field a jv and varsity baseball team and a swim team and a tennis team and everything else. The phenom would be playing alongside a novice. For that reason, the phenom would more likely go to a day school and play club baseball.
That small school dynamic exists on the East Coast, too, of course. My guess is that Cal not having PG students contributes to a comparative de-emphasis on sports, though.
The application process is similar, but getting into a competitive prep school in NYC in 9th grade is a lot harder than getting into a competitive boarding school. Mostly because they are all K-12 schools and the spots available in 9th grade are quite limited, Trinity is the only prep school that significantly expands in HS. The single sex schools in particular barely expand so spots are based largely on attrition. And they really do not care much about sports, with the exception of Poly Prep and maybe the hill schools. There are certainly kids that pursue competitive sports and get recruited (mostly D3 but some D1) but that is due to the club and out of school team participation for most part. The school teams vary greatly, depending mostly on how many kids are on travel teams. Soccer is decent, basketball not so much, and football is not competitive at all if it even exists. Catholic schools care more, recruit and have better teams, but mostly out in the burbs not so much in the city itself. In boarding school world the sports coaches have more say and some schools do recruit, though most of the high impact recruits come later than 9th grade. And since they start in HS, there are way more spots to begin with. I do think the line between prep and boarding schools is becoming more blurry as more boarding schools are becoming more like prep schools - getting rid of Saturday classes, increasing the number of day students, and having more students (both day and boarding) who leave campus on weekends and sometimes also weeknights to pursue their extracurricular activities, so continue to have lives outside of boarding school. This is both good and bad depending on whom you ask. Covid has put a pause to it temporarily at least. But as someone already said, if you want to play in college you have to play outside of the BS team because boarding school season is less than 3 months long and even if the coach puts together off season practices once or twice a week it is not enough.
I’m following closely since we are hoping our hook for BS is an impact athlete.
My kid is applying to BS and to a local day school. She contacted coaches at every school where she is applying. The BS coaches all zoomed with her. The day school coach came and watched (and freaked kid out because afterwards coach introduced herself to kid. I hadn’t warned kid that coach was going to watch because I didn’t want her to be nervous!)
She has another sport where she would make Varsity as a freshman, but is not expected to make a significant difference to the team. Some coaches zoomed with her, others just sent her contact details of students on the team and let her get on with it.
We only applied to schools with strong teams and coaching in her sport.
It’ll be interesting to see how the cards fall on M10.
Note that we are not in NYC so not dealing with those dynamics.