It really does depend on your area, some are controlled district wide and some are left up to the coach, and anywhere in between for others (AD, Principal, etc). I personally don't do any schools but have looked into it a few times trying to move that way since need something to do in the off-season. Especially while coaching I got to see how the schools I worked at handled it and asked a lot of coaches I knew how their districts went through the process. Always a wide range of answers.
Around here it seems to run two ways most...
A) The same company is hired throughout, they do the yearbook photos (head shots) as well as all portraits for the sports teams. They don't do any action shots and basically have drones with cameras doing the portraits - very basic, auto everything, quality sucks. They do only the head shots and portraits, group shots for some but not all. School is paying them to come out as well as parents buying packages. Some might work on commission of those packages alone but most I talked to said they "thought" the school paid them day fees and everything.
B) It's more independent and left up to the AD. They come in and do the T&I and give the yearbook copies of the team photo and maybe small group photos like seniors, juniors, captains, etc. Parents are paying for portrait packages and believe at some the schools are also paying for them to come out as mentioned, which you have to do because you don't know if the parents are going to buy enough to make it worthwhile.
High school T&I does not sell even close to as well as youth T&I...actually just had that conversation with some parents at the local school who had me do their youth wrestling T&I - the whole criteria was to have them "not fuzzy" or "blurry" like the high school photographer. Apparently the school photographer they hire has problems with having the images in focus and from the ones I've seen I can see why there's been complaints, they are far from...good.
Lastly, not sure if other states have this but in NY you have to go through a background check, fingerprinting, etc. to work in schools in any capacity. This includes the photographers and all assistants. Usually the company you work for is doing that but if you're independent you have to do it on your own.