Here's what we saw/did last year...
As noted there is no definite "scheduled" down time between Graduation and a B-split's report time/orders to their ship; some may well have orders and know where and when they are/will report to the ship before graduation; others may not get their assignment for a week or so after graduation.
Also as noted, KP will only pay for travel from KP (New York) to the ship or reimburse the midshipman for that amount they would have paid for the Government airfare to the ship.
Based on DS's experience last year my 2 cents are, if you and your family can afford it, it's worth it for your child to come home and spend the time at home rather than at KP, if they don't/aren't leaving for their ship virtually immediately after graduation. That said, it will likely cost you/them a few bucks as you won't likely get a flight for the amount that KP can get them for. In our case we were "lucky" in that we live near Dulles Airport and son needed to fly out to LA (Long Beach) to catch his ship. Thanks to cheap flights from DC (IAD) as well as from NY (JFK) to LA/Long Beach via Jet Blue, the prices for DS' flight out and back were very, very close to what KP would reimburse him for, at least what he was told the ATR approved as I'm not sure he's recieved reimbursement yet.
DS followed what he was told by the ATRs during the prep course and discussed and got all plans approved/coordinated with the ATRreceived before he made/executed them - I'd suggest that was key....