USNA has about 170 people on the football team. Virtually all of those were 'recruited' Most will never dress for a game in 4 years should they last that long and most are done NLT their sophomore year.
In the summer, there is time for intramural sports as well as USMA team sports including football. If you are trying out, you can go to your sport's practice and a coach may or may not notice you. At USNA, the final couple of weeks of Plebe Summer - the football players are separated from their Plebe Company and move in with the football team that is preparing for the upcoming season. I would think that requires an invitation.
For the class of 2020 at USNA - only one walk-on was invited to that session and he ultimately 'made the team' but never dressed. This same player was also recruited by a number of other D1 programs that had outstanding academics. This particular player would have been 'recruited' but since he didn't get his DoDMeRB waiver until very late in the process, the coaches had long since backed off his recruitment. So the coaching staff new his name b4 hand.
Many moons ago at USAFA, I think 25% of my class 'went out for football', none that weren't recruited were still practicing by Labor Day.
If you want it - go for it. USMA has a process. But be realistic and consider how many other D1 programs expressed interest in you and how that stacks up to USMA football. Plus, what makes you special enough to get noticed? Are you incredibly fast, are you particularly large framed where the coach can see that they can put 35 lbs on you and make a lineman out of you eventually.
And if you aren't tall enough, fast enough, quick enough for D1 football, USMA has some great club sports like rugby where many football skills translate. Plus there is also Sprint Football which isn't quite on par with D1, but appears to be a lot of fun and you get to play other schools vs intramurals.
If you have reached out multiple times with no feedback, you probably already know how interested the coaches are.
My experience with college recruiting is most coaches will have at least someone return the call even if all they can do is call your high school coach. (NCAA puts a lot of rules around coaches calling players back, but they call your coach directly)
It's a longshot, but you are smart enough to get into USMA and you appear determined to give football a shot. Strap on the pads and give it a shot - best to fight and die in the arena vs listen to us tell you it's not worth the effort and wonder later on down the line.