This is so “it depends.” For athletes who are targeting academically tippy tops, coming from a BS is a way of ensuring they have the academic “cred” for that. It isn’t unusual for an athlete to do a year of PG (usually after LPS) to shore up on academics for that reason. (They also do it to get another year of growth and strength.) So yes, coming from BS helps in this regard.
After that, it’s really sport dependent. It’s hard for a kid to distinguish himself in football, for example, at most BS. Soccer coaches generally favor kids from elite club teams, not just because of the level of play but for the ease of seeing them at big tournaments. Otoh, BS offer sports like crew and squash that are more niche. Timed sports (swimming, track) are some of the simpler ones for recruiting because a coach doesn’t have to see you play.
In many sports, the focus and travel to be a viable D1 recruit (so scholarship money) is at odds with the BS experience and the BS kids who pull it off are often day students or local boarders whose parents are making extraordinary efforts to make that happen. Not always, but often. Ime, these folks have chosen a prep school with recruitment in mind, as you suggest. And many BS are happy to have these kids for a whole host of reasons (including college matriculations that look great to prospective parents.)
For D3 sports, that still may be the case, but to a lesser extent. If you look at the bios on college sports rosters for the schools and sports that interest you, you’ll get a feel for where kids came from.
At our school, most (not all) of the recruited athletes are going to D3 schools where they can continue to play at a high, but not highest, level without having to make academic sacrifices to do so. But yes, that hook helps!
Personally, I think most day schools are set up to be (and are) more accommodating of families who are prioritizing sports for recruiting but still want solid academics. But that isn’t an option for everyone. And not what you asked.
Anecdotally, DS really got great coaching at BS and really blossomed as a player as a result - very much beyond expectation (recruitment wasn’t a thought when he applied)- and did end up on the radar of coaches. He was not at the level that he had his pick of schools but he had offers of roster spots at several he applied to. All the college coaches asked if they could talk to his BS coach who agreed and was helpful.
This ended up being so long! But in a nutshell, I would say that if it’s a top D1 program you are targeting in a non niche sport, BS is probably not your best choice. If it’s a selective D3 school (i.e., NESCAC), odds are better.