I think how challenging admissions is for day students likely varies dramatically from school to school. Location (other boarding schools nearby competing for the same students, area population density, quality of local public schools and day schools), the proportion of day students the school is willing to have, and day students’ role in school culture probably all have a big role. There are schools where the % of slots that will go to day students who aren’t fac-brats, legacies, or recruitable athletes is in the low single digits and schools where it’s half of the student body. Schools near major urban areas are drawing from different day-student applicant pools than more remote ones. And so on.
Anyways, the question about a boarding requirement brings up a good point about Hill school - they do require all students to board for at least a year (usually senior/6th form). Some students and parents really like this as a sort of transition year before college, and by that point most day students are so well integrated and comfortable in the school that it doesn’t feel like a huge leap the way going off as a new student boarder freshman year. But it’s definitely something to be aware of. Hill also has a more conservative, traditional culture than average.
If I remember right, Hill no longer has Saturday morning classes but I believe Peddie and Lawrenceville do. That’s common at boarding schools so definitely something to check about anywhere he’s considering, especially if the commute will be long.