Let me offer a ray of hope along with a little personal experience to the OP.
I've been shooting sports for 40+ years. Started in high school, shooting from my school. Getting into games was a breeze. I also was a high school kid with a high school kid's mentality. Got some good shots, but it was a bit hit and miss.
After high school I got a lot more serious. I shot for various entities, publications etc. Yes, that did make it easier. Keep in mind photography was not my main employment. I enjoyed photography too much to want to ruin it by making it my job. It was here that I truly learned the craft of Sports Photography.
When our kids were born and being raised I did a lot less. As they grew I got back into it shoot them and their teams. As they were getting older, going off to college getting married etc. I ended up with more free time on my hands. Two of the kids were okay athletes. the middle son however was a good track and cross country runner. He received scholarship offers from several schools and choose to go to one of them and run. Fortunately, it was a home grown college so shooting cross country was a breeze. I talked to the AD at his college and after discussing it for a while he let me shoot any of the sports I wanted from the sidelines/court side. He realized I knew the craft and knew what to do and what not to do.
The cross country coach believed in challenging his kids every year with at least one really big meet. A Division 1 meet with teams ranging from the smallest of private colleges to programs like Arkansas, Stanford, Arizona, Colorado etc. and everything in between. They were in between. The meet was only a couple hours away so I took a day off from work, went and shot the meet. While I was doing so, I watched a lot of Parents with cameras all fumbling around trying to get a shot of THEIR KID. Me, I didn't even pay attention to the fact that my son was running in the meet. I was there shooting sports. Because of this, I was one of the few that knew what needed to be done, how to do it and how to stay the hell out of the way.
As the girls race was getting near the end I headed toward the finish area. There was a mass of people in the spectator area and it was a poor place to shoot from. This meet was being handled by a professional crew who had brought in their own timing equipment as well as finish line chutes etc. The perfect place to get the shots I wanted of the finishers and the race to the finish was from this crews side. I politely explained to the person who I was and asked if I could shoot from their side of the finish. The person I talked to had no problem as long as I stayed in a particular area which would not interfere with his crew getting the finishers in proper order through the chute. I readily agreed.
While I was shooting the finish, another photographer comes up next to me. We exchanged greetings, worked to stay out of each others way and talked a bit between the girls race and the boys race. Nice guy.
As I was packing up to leave after the races were over a cushman pulls up next to my vehicle and the driver is the same guy I was shooting with who happened to turn out to be the Assistant AD in charge of publicity for this very large University putting on the meet. Well Shut My Mouth.
He asked who I was shooting for believing I was there for one publication or another and I explained to him I was just shooting for me and the kids at the school where my son ran. He was surprised and expressed it telling me that my attitude, knowledge etc. was very professional and he thought I was there for a publication. We talked a bit and long story short, he offered to let me shoot at any and all sporting events at that University. Well needless to say, I took him up on his offer and have shot some of the best football, baseball, and other sports that University has to offer, and they have a lot to offer in the Collegiate sporting world.
He also offered to call other AD's or the appropriate Assistant AD's at all the other Big 12 Universities if I wanted to shoot at other Division I events. Not one of those schools ever turned me down. I have shot and continue to shoot field side and court side for some of the best teams and best College sporting events that the NCAA has to offer over the last 8-10 years.
There are some limitations. I can't shoot certain championship games or meets and it is a bit tougher shooting at the "neutral site" games as they are generally held in places like Chiefs Stadium, Cowboys Stadium, etc. but hey, I don't mind. I shoot for me, not a publication so to me it is no big deal.
If you want the opportunity to shoot something besides high school sports then here is my advise. First, start shooting at the local high school. Talk to the person in charge of athletics there and explain to them what you would like to do and what requirements they have to let you. From there, work very hard to learn your craft. That is not easy. It isn't just about taking pictures. You need to learn the ins and outs of every sport you intend to shoot. Not just the rules, everything about the game, including what and how the individual officials will be working the game. You need to learn the players, their strengths and weaknesses. You need to learn your gear and buy or rent the best gear you can afford for the various sports you want to shoot. Finally you need to learn how to get the shots you want with out impacting the event in any way shape or form. IE: you need to learn to be a professional.
Once you accomplish that then call the AD or Assistant AD at the local college or university and speak to them. Take some of your work to show them. The worst that can happen is that they will tell you no.
Keep in mind, if you are serous about this, it is hard to look like a MWAC and end up getting permission to shoot at these kinds of events. The basic tools of the trade for me these days is a 1 Dx, a 1D MkIV, lots of fast glass for indoor, and outdoor. That includes a 200mm f2, a 300mm f2.8 and a 400mm f2.8. All expensive glass, but if you plan right and watch around you can get some great glass at a good price. My 400mm I bought new. The 300mm I got slightly used from a local doctor who bought it when his daughter was in high school. Once she graduated he quit taking pictures at the sporting events at the high school she had attended and decided to sell it. The 200mm I got at a local camera store for about the same reason. Again used, but in excellent, hardly used shape for about 1/3 of what it cost new. I still have a chunk of change wrapped up in what I use, but it could have been worse.