Boxing in the ancient world was much simpler than what we practice, so in a lot of ways the specifics of your question don't make sense when we realize what was actually going on in a boxing match. Greek boxers stood (originally they may have sat) in a very small circle and they just traded blows over and over until one gave up or was incapacitated. That's it. We have no indication of any sort of real blocking, either in art or literature, and movement is completely absent in all depictions or descriptions of boxing we have, unlike wrestling or the pankration in which our descriptions are always talking about the movement of the fighters in the ring and their maneuvering around each other. Boxers did not literally trade blows on after another, and complex combinations of jabs and punches were common, but blocking doesn't seem to have really happened. At most you dodged your head or punched the guy first--the latter is well-attested in vase paintings, where a common theme is a boxer hitting his opponent mid-strike. The point about dodging with your head is important too--boxers hit each other in the face, and body blows did not happen. We don't know whether body blows were illegal (it seems highly unlikely) but they were rare, as the orthodox practice was to attack the face. A Greek boxer would have found our practice of fighters being allowed large rings in which to maneuver and run away from each other, as well as the practice of regular breaks in the fighting somewhat silly.

The bottom line is that boxing was a test of pure strength. Athletic competitions in Greece were not like ours in origin. Our athletic competitions are designed partly for spectacle and partly for the benefit of refining individual athletes' technique. Although athletic competitions in the ancient world ultimately derived from tests of martial ability, the organized athletic events of the Greeks derived from festivals and funeral games. It's quite apparent from the Homeric description of Patroclus' funeral games and later descriptions of heroic athletic games that athletic contest was a sort of heroic contest, in which athletes could compete to prove their individual prowess. The origin of the rules and organization of these events is not to provide athletic training nor to provide recreation, but, at least in the minds of the Greeks, to allow for heroic competition. And, as we can see from the works of poets like Pindar, the winners of athletic competitions were viewed as heroes, receiving honors parallel to those of the heroes worshiped in hero cults. At one time scholars proposed that funeral games developed into a form of heroic competition that allowed everyone, not just the elite to compete and to enter into hero status--I'm not sure how well-received this model is, as this is far from my specialty within Classics, although I rather doubt that many agree with much in the assessment. Still, the heroic aspect of athletic competition should not be overlooked--boxing was a way for an athlete to demonstrate a particular skill, namely his pure strength and force of will. Wrestling and the pankration had a great deal of finesse, skill, and strategy involved (particularly the pankration, in which often the weaker fighter would win, sometimes by resorting to things like breaking the bones of his opponent's extremities and wearing him out until he gave up due to pain). Boxing did not, it was all about proving who had more raw strength

EDIT: I should add (and this is important, I don't know why I forgot it) that the pankration was a late addition to the Olympic repertoire, and most of the early games were very simple and required very little skill. Our modern sporting events require both pure physical ability and the skill and experience in the sport to master the tactics and nuances of winning. A modern boxer or wrestler doesn't go out there with the intention of throwing all his power willy-nilly into the ring expecting to win. Instead, he knows that he has to be able to actually fight with his opponent, maneuvering, feinting, advancing, retreating, etc. Most of the earliest Greek games did not have this aspect, or restricted it to a relatively minor role. The footraces were originally all sprints, the idea being that you throw all your physical strength as a runner into the race and either you win or you don't (or course, as a runner myself, I know that it's far more complicated than this, and even Homer is aware of that--which may be a reason why the Greeks prized the footrace so much). The throwing events required little skill besides the simple mastery of the actual technique used to throw--you hurled your javelin or threw your discus and either you had enough strength to throw it farther than the other guy or not. Boxing is the same--either you have the power to kill or incapacitate your opponent, or at least beat him to a pulp so that he gives up, or you don't. Of the earliest games only wrestling really stands out, as it requires a lot of skill and technique to know how to grapple your opponent and pin him to the ground. In Homer it's notable that Odysseus fights Ajax during the wrestling match of Patroclus' funeral games. Unlike the boxing match, where Euryalus tries to punch Epeius first but is knocked off his feet by a single punch (and is commended for daring to go up against Epeius, the champion boxer, anyway), the wrestling match is decided by something other than pure strength. Ajax, the strongest of the Achaeans after Achilles, is knocked down when Odysseus sweeps his leg (although the match is declared a draw, as Odysseus cannot pin him), whereas Ajax attempts to win by the usual tactic of just grappling his opponent and pushing him down. Wrestling is something of an anomaly for the early games, and the pankration certainly would not have fit in if it had been introduced at the earliest period

Unarmed melee combat can be roughly broken into 3 discrete categories.

Strike: Typically performed standing. Keeps a moderate distance between opponents. Distance is closed only to "strike" the opponent, then the distance is reopened. Uses various parts of the body similarly to how weapons are used.

Clinch: Similar to strike in that it is typically performed standing and damage to the opponent is typically performed through striking blows. This technique differs from strike in that grappling techniques are used to ensure that the opponent never gets out of very-close-range striking distance, and can be used to neutralize range advantages. This is a very close quarter combat and is primarily domination based. Often it is used to transition from striking to grappling via take-downs, sweeps, and throws.

Grapple: Focused on holds, bars, and chokes. This technique puts combatants in extremely close quarters. Any strikes performed are typically accessories to grappling.

Most forms of martial arts will employ a variety of techniques from each of these 3 forms of melee combat. However, they tend to emphasize one of them. As for the Greeks and Romans, their martial arts focused around their battle skills. Greek and Romans viewed grappling and striking separately. This is likely why we have Greco-Roman wrestling and Boxing.

Obviously, wrestling has its own merits, but the strength of an individual plays a large role in his ability to dominate his opponent. Not all men are strong or large, so we rely on other schools of combat as well. Namely, strike fighting, which relies on agility, speed, and technique. Notably, strike fighting also avoids the knife in the kidney vulnerability, which is every present when grappling, so it is useful to everyone.

Most Greek and Roman competitions were created to simulate the physical strain two competitors would face in combat. Obviously two friends wouldn't want to kill each other to find out who was stronger, so we devised a variety of games. Boxing is amongst them and is notable for a variety of reasons, but we'll focus on what I believe to be the most notable.

Boxing accurately recreates the types of combat two opponents would likely face in a battle. The Greeks and Romans relied heavily on artillery during the first phase of battle; however, most soldiers would experience melee combat as infantrymen. Their weapons would be spear and short sword (gladius) both of which were piercing weapons used with thrusting or punching motions to penetrate armor. They could be swung in order to slice or stun, but their main advantage was piercing.

Boxing uses the same kind of piercing or thrusting strike techniques and also tests an individual's ability to evade the same. In this way, young men could hone skills that would easily translate to the battlefield without risking serious injury. Moreover, it's entertaining and allows for friendly competition.

It is worth noting that boxing has never been used as the sole form of martial training for any successful fighting force. It has only ever been an auxiliary training technique and a game.

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[Edit] Above I've seen people discuss boxing's role in building a soldier's pain tolerance and stamina. This is simply not the case for the Greco-Roman world. Soldiers trained stamina by marching, sparring with dummy weapons (twice the weight of real weapons), and by exercising in a gymnasium. (Source: http://www.roman-empire.net/army/training.html)

Pain tolerance was also trained in fairly brutal manners. They would beat recruits, whip them for disobedience, force them to march, and there are other accounts of training pain tolerance. One famous (hearsay) technique used by Romans for increasing pain tolerance involved holding ones hands to a flame for a longer duration each day. Once captured, Roman prisoners would demonstrate the futility of torturing them by holding their hands in fire (Source: http://en.wikipedia.org/wiki/Gaius_Mucius_Scaevola) Apparently the bit about fire is bullshit.

Sources:

Source 1: http://penelope.uchicago.edu/Thayer/E/Roman/Texts/secondary/SMIGRA*/Pugilatus.html

Source 2: http://www.tribunesandtriumphs.org/roman-life/roman-sports.htm

Source 3: http://en.wikipedia.org/wiki/Roman_military_personal_equipment#Personal_weapons

I'm currently reading about a Victorian mixed martial art called Bartitsu and that literature discusses boxing and folk wrestling in England.

Originally boxing was brutal and included kicking, grappling, eye gouging, and other ways to maim your opponent.

From boxing historian Nat Fleischer, regarding boxing in the 18th century, "It was considered manly and proper to kick a man when he was down ... and eye-gouging was a common practice with the fighters and hugely enjoyed by thr spectators. The use of fists was relatively incidental."

John Broughton then introduced some rules to boxing sometime in the mid-18th century, though I don't have an exact date. These were expanded on by the London Prize Ring in 1838. During this period boxing was still bare-knuckle, but the 1838 rules update prohibited fouls such as biting or head-butting.

The next major rules update came in 1867 with the Marquis of Queensberry rules. These superceded the earlier London Prize Ring update of 1838. The Queensberry rules finally required that gloves be worn and outlawed wrestling and grappling. These rules caught on, but bareknuckle prize fighting under the London Prize Ring rules persisted for another 30 years. Only as amateur boxing and the physical culture movement caught on in the late 19th century did the Marquis of Queensberry rules push out the last remnants of the London Prize Ring.

Source: The Bartitsu Compendium, Volume 1, edited by Tony Wolf.

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To address the another part of your question:

"Did they see a value in punching that is not given by wrestling, for example in soldier training? This would be my most important question. Why would the apparently least efficient form of unarmed attack (punching) be advantaged by banning other attacks (kicking, grappling), what is the point of promoting punching? Does it have a different value, like, helping in later on in armed fighting, with swords or spears, or, helping in things like enduring pain?"

From studying various treatises and manuals on individual combat in Europe from the 13th century through to WWI, restling with an opponent on the battlefield may form a framework for understanding the material that follows, but the impression that I am always given from it is that wrestling and especially ground fighting is a bad idea. You need to have an understanding of it not to use it on someone, except to take them to the ground and finish them off, but rather to be able to defend against it.

Rolling around on the ground exposes you to too many dangers not only from the compromised position but also to being a target for your opponent's allies.