Why is it that flying aces gain more recognition than other members of the armed forces?

(since I'm nearing the 10k limit, see my replies to this post for further info, where I will go into some further detail regarding specific aspects of your questions)

As /u/classicorange says, a lot of it is down to the romanticized nature of aerial warfare.

Aerial warfare more than most other forms of combat today relies heavily on individual skill. You also, as /u/classicorange says, have the fact that engagements tend to be much smaller scale and more "one on one" so to speak. Sure, during WWII especially, you had hundreds or even thousands of aircraft in the air in certain engagements, but even then the combat generally broke down into much smaller individual engagements composed of a few handfuls of aircraft at once. It's just not quite the same thing as having hundreds or thousands of men storming a defensive position or some such. An individual soldier on the ground could die for many, many, reasons beyond his control. He may be killed by "faceless" artillery, or a lucky bullet fired from an unskilled soldier blindly firing across a field. A pilot on the other hand, while still vulnerable to things like flak (the equivalent of enemy artillery in a ground engagement), still can overcome an enemy in a superior aircraft through proper training and personal skill. See my reply to this post for a copy-paste of another answer I gave in another thread regarding the "personalized" nature of aerial combat.

Kills are also much more easily identifiable and attributable to an individual pilot as opposed to combat on the ground. I think this is why snipers also tend to get more recognition than most other "grunts" on the ground, because they tend to be individually stalking their prey, and their kills are sometimes easily confirmed. See another reply to this post with another copy-paste of mine for how kills were verified, and also why confirmation of kills was important instead of just relying on a pilot's word for it.

As for why missiles don't matter much, well, they still require a lot of skill to use. Air to air missiles aren't the "point and click" kind of kill that they have the reputation of having. You still need to place your aircraft in an advantageous position, and even then there's no guarantee that the missile will find it's target. For example, in the Vietnam War, the American made AIM-7 Sparrow missile had a less than 10% success rate. That is due to a variety of reasons, sometimes the missiles failed to even ignite upon leaving the rails, other times they failed electrically/mechanically for other reasons such as failure to keep lock on the target, and other times simply because the opposing aircraft outmaneuvered the missile. Missiles have gotten much better since then, but they are still relatively easily defeated if you know one is headed your way. A common tactic taught today to pilots to defeat modern radar guided missiles is to simply place your flight path perpendicular to the flight of the missile (so 90 degrees or greater), and then simply maneuver (such as by turning into the attack) so as to overwhelm the missile's ability to maneuver. They simply can't keep up and turn tight enough to match such maneuvers.

Also, restrictive rules of engagement, such as seen in Vietnam, often mean that aircraft have to visually identify their target before engaging, which means you have to close the distance so that you're essentially dogfighting. Ranges might be a bit further than with guns, but you're still often going to find aircraft tangling with each other within visual range of each other. And again, you still have to put yourself in an advantageous position to engage. Modern missiles don't have quite the same "get on your enemy's six (directly behind them)" requirements as in the past, but it still is the best place from which to engage the enemy.

As for why the Germans, here is a thread that discusses why Germany has so many aces that dominate the list of aces.

Here's my copy pasted answer from that thread (though I recommend looking into that thread for a more indepth discussion):

There are a few factors for this.

One, and most notably for those aces that top the all time list, is that they very often flew until they died or were otherwise unable to continue flying. For example, the all time ace of aces, Erich Hartmann, first joined up in 1940. He was then posted to front line duty in October 1942. He flew until the end of the war, securing his final kill in early May, 1945. The Japanese, who also had a few pilots rack up a large number of kills, also often flew in such a manner. While this allowed them both to have pilots that attained extremely high numbers of kills, it also came back to bite them in the ass later in the war as more and more pilots were killed. Their ranks simply became thinned of experienced pilots to lead the less experienced ones into battle, and also to pass on their knowledge. The Allies on the other hand, especially the Brits and Americans, tended to rotate pilots back home to spread their knowledge and experience with the next round of trainees. So simply put, the Allied pilots just didn’t have a chance to rack up huge numbers.

Both the Japanese and Germans also had a technical superiority edge over their early opponents. The Germans for example, often came across obsolete Soviet aircraft manned by poorly trained pilots and poorly organized commands. The Germans also took home valuable lessons from their experience in aiding the Nationalists in the Spanish Civil War. It should also be noted that despite what you may hear about such vaunted aircraft as the P-51 supposedly being so superior to the German offerings, the Germans actually kept up pace relatively well with the Allies throughout the war in terms of fighter design. The two principle fighters of the Luftwaffe, the Bf-109 and Fw-190, were both constantly updated and improved to keep pace with fighter design improvements and more or less kept an even edge with their Allied counterparts. There was a bit of back and forth wherein one side might for a very short time attain a technical superiority over the other side, but such instances were always short lived. In any case, what it almost always came down to was pilot skill (in other words, any technical superiority would have been minimal and easily overcome by an experienced pilot fighting to his strengths and his opponents weaknesses).

Another aspect is simply how kills were tallied. The Germans had a “one pilot, one kill” system. This meant that only one pilot could claim the victory even if more than one pilot contributed to the kill. For various reasons, the more experienced pilot may be given credit for the kill. British and American pilots used a fractional system, wherein credit could be shared among pilots but the total would still equal one (so two pilots shooting down one aircraft would be awarded 0.5 of a kill, four pilots taking down an aircraft would each achieve 0.25 of a kill, etc). I will have to double check on the Soviets as it escapes me what system they used, but I’m pretty sure they, like the Japanese, awarded kills only for solo victories, and counted partial kills separately (though the Imperial Japanese Navy abandoned individual victories for those of squadron victories in I believe mid 1943). The French awarded one kill to each contributing pilot (so four pilots contributing to a kill would all receive a full credit for one kill. So you might have one downed aircraft, but four claimed kills by four different pilots). The Finns, who also have a few aces pretty high up on the list (and the highest ever non-German ace) also used the fractional system as the Brits and Americans. The Americans also, at times, counted aircraft destroyed on the ground as an aerial victory.

Another reason for German domination of the list of all time aces, with the vast majority of these coming in WWII, is that expert pilots often seemed to be clumped into the same squadrons. Part of this is simply due to the fact that expert pilots can go on to share their experience with lesser experienced pilots, who then learn from these teachings and go on to become experts themselves. Jagdgeschwader 52 for example, is the most successful fighter wing in the history of aviation, and counts the top three aces of all time as members of the unit. JG52 started off in the Battle of France and Battle of Britain, where it didn’t really stand out, but upon transfer to the Eastern Front, they started racking up kills. In the last 6 months of 1941, they racked up nearly 1000 kills to only about 50 losses. Six months later, they had racked up another 500 kills, and then in the period of a month in mid 1942 they had racked up another 500. 500 victories in a single month. All told, they claimed over 10,000 victories for a loss of roughly 750. Quite staggering really. Again, part of this is because of their technical superiority, but also experience. So as you can see, it all kind of ties together. Their technical superiority may have made survival more likely, leading to more experience, which would pay off as the Soviets came up to speed in both pilot training and equipment.

But if I had to pin it on one factor alone, it would simply be the duration of a pilot’s time on the front line. As I said before, most of the Allied pilots simply didn’t have the time to accumulate large numbers of kills.