Current poli sci and fly practitioner along with
@Ivy22 here. To answer your initial question, most of those who apply get a flight spot. However, year to year, it can vary. My year had a surplus of applicants. Two years before me, the amount of applicants matched up roughly to the slots available. Those who I saw get denied it were generally those who didn't have stellar statistics and also did not do much to demonstrate their interest.
As
@Heatherg21 said, networking is a huge part of Service Assignment. Granted, you have to hold up your end of the bargain: Do well in school, pass the PRT, don't get in trouble, etc. But, X factor is showing your interest in a community. Are you going to every fleet fly in? Do you schedule times to go meet with military staff/faculty and talk about their careers? All of them were operators in a past life. They are not career administrators or academics. Heck, even some of the senior officers move onto major command after their USNA tour. People will remember names and faces and it is more difficult to deny someone you know wants to be part of your community than someone you know nothing about. Also, people will sometimes shy away from the senior level (O-5/O-6) crowd. I have found they are actually the most excited to talk to MIDN. All of them are there because they want to develop the next generation (i.e. you). Also, talking about their career allows them to take a break from whatever staff/academic work they have on their plate.
Major is not an influence in Service Assignment outside of your QPR. What I mean by that is that your QPR goes into your Order of Merit and if you do a major you are good at (and hopefully you like), then it will bode well for your Order of Merit. There are myths that say that you have to be an Aerospace Engineering major to go Pilot or a Nuclear Engineering major to go Subs. Not true at all. I have seen English majors go Subs. I have seen Aerospace Engineering majors go SWO.
In addition to Navy Pilot, I would also do a little exploring into USMC Pilot and NFO. Both provide flying opportunities to Navy Pilot. NFO has gotten more competitive as it has gotten a "face lift" at USNA. It very much had the "not good enough to be a pilot" connotation when I arrived. Now, it it is very much advertised (as it should) as a very integral part of Naval Aviation. They are the weapons/sensor employment experts. Depending on the platform, pilots are flying as a means to allow NFOs to do their work. USMC Pilot offers another chance to get a pilot slot, but they are very selective in who they choose. The main difference between the USN and USMC is how they approach Service Assignment. The USN is very focused on different communities (SWO, Subs, Aviation, etc.). The USMC is very focused on picking good future Marine Officers with less of an emphasis on their specialization. In other words, you have to be all in being a Marine rather than being a Pilot.