I'm not in the governance of interscholastic sports for youth, but I can see why interscholastic federations would be hesitant to allow active student athletes to receive coaching from someone who is outside the "circle" of interscholastic sports.
First and foremost, student athlete safety is in question. I'm pretty certain most private coaches are decent people, but most are not likely be cleared with Department of Justice as individuals approved to work with your children.
Second of all, I think that particular rule is intended to even the playing field. In many interscholastic federations, and especially in high school districts, there are almost always huge discreptancies with resources that are made available to each school.
Thirdly, I think forbidding students to work with coaching outside of what the school provides could be a way to protect the integrity of hired school coaches. We all hear about coaches that complain about individuals who are not on the coaching staff (ahem, crazy dad) attempting to tamper with the athletes, while in competition and practice, which that outside instruction could be outside the boundaries and philosophy of what the coach wants to do.
In the end, interscholastic bowling in the K-12 arena is in a strange position because participation largely takes place on private property. Additionally, a large amount (if not all) of the activities, services, and products are supplied by private businesses and acquired by the kids/parents themselves. Rarely do the schools provide a lot of the supplies, outside of lineage.
The biggest can of worms that has been opened up from this discussion is equity and standardization related the playing conditions and equipment available to athletes. It is no question there are certain demographics that will have more access to services and equipment than others. If you think about it, many of the other interscholastic sports available do not have a lot of variables from equipment and resources available that affect the game as much as bowling does. For example, in interscholastic soccer, all that's required is a ball, cleats, a safe playing field, "qualified referees," and nets. All of those resources are easy to get and are fairly standardized. In bowling, the equipment necessary is expensive, and it is clear that the quality of resources available to you play a big role with success.
If you asked me, if interscholastic bowling would be accepted as a legitimate interscholastic sport, the equipment and playing conditions HAVE to be standardized to force the athletes to rely on skill as opposed to the resources available to them. Interscholastic federations have the power to do this. The best example of this is the Teenmasters model, where a standardized bowling ball and playing conditions are pre-established before competitive "season."
Bowling as an interscholastic sport is far from perfect, and right now, all I would care about is that the kids are bowling. It beats no bowling at all.
Finally, if the interscholastic bowling scene doesn't work out, parents and student athletes have a choice. They have the "club sport" route. Examples of this are: Junior Gold/USBC, JAT, Youth Bowling Tour, and JBT. All of which are great competitive venues for youth to develop their games, if you have the resources!
Just my thoughts.