I was a Boy Scout as a boy, am currently registered as a Boy Scout leader, and it's been over 10 years since my oldest child started Cub Scouts. As mentioned in a comment, the form you're asking about is probably the Annual Health and Medical Record, parts A and B. (There's also a parts A, B and C version of the form that includes a page that must be filled out by a doctor.)
My understanding of the way the form is used is that it has two purposes:
- It allows event organizers and supervisors to take reasonable precautions to keep all participants healthy (not just the adults filling out the form for themselves, or children whose parents filled out the form for them, but also other children whose parents entrusted them to the organizers of the activity).
- In the event that a participant is injured and requires medical care, and is unconscious or incoherent, the form will be shared with first responders or the Emergency Room. (Although I've heard from leaders who actually had to take someone to the emergency room that there was little opportunity to introduce the form, even though they had it ready!)
It's not used to deny someone membership in Boy Scouts of America (the youth or adult membership application is a separate form, and has no medical questions.) And while various people might possess the form and many of them are unpaid volunteers, the tradition is that the form is kept private and only shared with those who "need to know" its contents. At day-camps and similar activities run by my Council, the form is reviewed once by a designated health officer at check-in, kept in a secure place by that officer, and returned to an adult participant, or the parent or other leader of a youth, at the end of the activity.
And if the information disclosed on the form suggests that the subject's participation is unsafe, that might mean the subject is refused participation in the activity, but it might also give the organizers an opportunity to prepare an accommodation. For example, suppose there is a ladder on a certain course that is rated at 200 pounds, and you report that you weigh 215 pounds. They could tell you that you can't go on the course, or they could replace that ladder with a ladder rated at 250 pounds.
(Although for a situation like that, it might be prudent for the operators to have a scale at the course entrance, and re-weigh everyone before entry!)
While I've dealt with private tour operators and standalone youth camps that had shorter liability-release forms, that might not be the best comparison. Typically a tour operator runs one specific activity (whitewater rafting on a particular river, or climbing a particular mountain). The BSA program could include activities all the way from playing with dough (could trigger a gluten allergy in a child who nibbles it) to scuba diving, sometimes at an organized "camp" like the one you were going to and sometimes at public parks or even privately-owned backcountry.
A better comparison might be the Utah High School Athletic Association's Pre-Participation Examination Form, or "Physical Form A". Not counting the instructions, that's three pages. Utah isn't alone in having such a form for student athletes; I live in Michigan, and have had to fill out the Michigan High School Athletic Association's Physical Card/Medical History Form every year I wanted my children to play sports.
And just because another organization's form is shorter doesn't mean that it doesn't require medical details. For example, one well-known church headquartered in Utah uses a Permission and Medical Release Form (description), which includes questions under the headings "Medical Information", "Conditions that Limit Activity", and "Other Accommodations or Special Needs". The instructions for that last block end with "(attach additional pages if needed)."