Some truly gobsmacking tosh there.
My DCs were registered with the practice of the hospital pediatrician who saw them from birth. They saw the pediatrician from birth to age 18. He was great, very supportive of breastfeeding.
Initial visits were 'well child' visits, beginning with a two week checkup, and at these there was weighing, measuring, and beginning at two months the mandated vaccination schedule kicked in. Iirc, they went at 2, 4, and 6 months, and again at 12 months, for their growth and development check and for shots/ boosters.
After that, the schedule of visits followed the vaccination schedule until maybe age 4, when annual checkups became the norm. Proof of an annual physical exam is mandated by the state I live in, along with proof of an annual dental exam, in order for a child to attend school. Proof of vaccinations is also required.
My DCs did their annual exams every summer all through school as high school required this too. They played sports in high school and the state schools' athletic board required a sports physical - the sports physical sufficed for the school/ state requirement iirc. They got some booster shots in their teens, gardasil, and also meningitis shots in their late teens (required by universities).
The pediatrician we were registered with was a professor of pediatrics at a local medical school. His practice saw a huge number of children and we always had residents ( qualified doctors learning their specialty) at the visits, doing the physical exams, and being questioned by the pediatrician.
There were experienced pediatric nurses at the practice who fielded phone calls and liaised with the doctor, if something came up. They dished out advice over the phone and basically did phone triage, with the aim of keeping office visits to a minimum.
We had an acute asthma attack and a few ear infections that needed same day attention over the years. Plus one occasion when toddler DS stuck a staple into an electrical outlet a week after we moved into our house, and got an electric shock. There were many years when we only darkened the door of the pediatrician once, for the annual checkup.
Pediatrics is a primary care specialty in the US, as is OB/gynecology.
There are also family practitioners, who see people at all ages and stages of life. Managed care health systems tend to employ family med providers, who refer to specialists as needed.
I have a family practice physician at this point of my life - she referred me for a battery of recommended tests last year (colonoscopy, mammogram) and all follow up testing and procedures were ordered by her too. She also ordered blood tests for various menopause related symptoms. The office phlebotomist did the blood draw and the sample was sent to a lab. My doctor does my annual pap smear herself.