*sigh* so many flaws in reasoning...
Despite being a crappy one, Shirou is still a mage, and as such, is fully capable of summoning a servant. Artifacts can help determine your servant, thus any mage can summon them regardless of skill. However, as is the case with Shirou, when a lousy mage summons them, their stats are lowered. Arturia meets all the stat requirements for the Saber class, but her stats are lowered because of Shirou's lack of ability as a mage.
Umm, no, Archer is dead. He was betrayed and killed by someone that he saved. Arturia is the only Servant that became an Eirei whilest still alive.
Rider was not "summon jacked", Shinji just has Sakura afraid of him psychologically, despite him not having any powers. Rider was willingly given by Sakura to Shinji, I believe from using a command spell to make the book that Shinji carries that gives him control of Rider as her master.
Again, one exception does not make it a fact that the rule is easily broken. Assassin is a bit messed up from being summoned by a servant. True Assassin is around and fits the bill of the typical Assassin class.
Caster was summoned by a mage, whom she promptly killed. She was then found by a non-mage, and decided to stay with him, and have him act as her master. He is more like a partner then an actual master.
Avenger is a legitimate class made by the Einzberns with the intent of summoning Angra Mainyu. However, since Angra Mainyu is full divinity, it did not work. The current Avenger was not what they intended, only a result from the fact that they could not summon what they intended. He was the closest alternative.
The only person who actually summoned a servant that can really be considered unskilled as a mage is mainly Shirou. Rin and Bazette are skilled mages. If I recall correctly, Sakura has access to a large amount of prana, and Ilya has a large number of magical circuits, if not that she was almost entirely made of magical circuits. Caster's summoner...who knows, he was killed as soon as he summoned her. And of course, Caster is obviously a skilled mage.
In addition, aside from Assassin, no one was summoned to a class to which they don't fit. Lancer has the high agility and close combat skills for his class. Arturia has the stats for the Saber class, only that they are lowered due to Shirou's lack of talent as a mage. Caster has the MGI stat to fit her class, Herakles went insane at one point in his life. Assassin is the only one who does not fit, though True Assassin does.
Rules for classes are not loose guidelines. They are rules. The only exception is 5th war Assassin, and that is one exception, not a standard. Stats are a definitive description of ability in terms of what class they can and can not fit into. However, their qualifications are based on their base stats, not on their stats after summoning. The skill of the summoner does not determine who they get, but the level of their stats.
Again, one exception does not set a standard. It is one instance of the class being broken, and only when the summoning is done in a non traditional way. It can not be held as proof that the rules are always breakable when it is only one occurence.
Unskilled masters can summon "high level" servants, but with lower stats, thus making them weaker. Summoning of servants is done mostly through the Holy Grail, and as such, does not require much talent as a mage to summon. Skill level does not make a difference for who you summon.
Saber: Summoned by pseudo-magi, on accident, because Shirou happened to have an artifact associated with her.
Archer: Not actually dead. Also from a parallel universe.
Rider: Regardless of what we mean by "hero," she wasn't even a human. Anyway, she was summon-jacked by a kid who didn't have any actual magic power.
Assassin: Not a Hassin. Summoned by another Servant. Possibly not a Real Boy.
Caster: Summoned by non-magician
Avenger: A completely made up class for summoning a peasant from way back in the day that was a human sacrifice. (Maybe he wasn't in the game but a sequel is close enough).
So in the fifth war, we see a lot of unskilled summoners calling forth Servants who can't technically be summoned into their class.
In addition, aside from Assassin, no one was summoned to a class to which they don't fit. Lancer has the high agility and close combat skills for his class. Arturia has the stats for the Saber class, only that they are lowered due to Shirou's lack of talent as a mage. Caster has the MGI stat to fit her class, Herakles went insane at one point in his life. Assassin is the only one who does not fit, though True Assassin does.
I tend to the line of thought that these "rules" are at best loose guidelines, and that as long as someone can be kind of matched with the corresponding servant motif, it's all good. Plus "stats" are a pretty crappy way to actually describe someone's ability, especially since they also seem to be dependent on how well they were summoned or how skilled their summoner was and stuff.
For example, everyone talks about how summons have to be Hassins. Well, okay, but the Assassin summoned for the Fifth wasn't a Hassin. Even if Medea was a super-sorceress, it's still a breakable rule. That seems to be a theme: yeah there are rules for who can be summoned where, but those rules are breakable.
Also we see unskilled masters summoning high level-servants; I don't care how much of a prodigy they are, I just don't think a high-school student would have enough job experience to really be a skilled Magus.