I don't necessarily agree or disagree with the design given as it was the first of it's kind and created in a different era. But here is my understanding of it with some lead in for context:
The Ability Scores represented six characteristics of real world adult humans and were not really meant as character attributes to be portrayed. Their generation followed a bell curve meant to represent our world's demographic distribution of these six abilities among the adult human population, IIRC for ages 16-50. The numbers represented 16 segments within that distribution. (Demographic segments are vertical slices of the bell curve, sometimes demographers use only four calling them quadrants) So these 16 numbers, listed 3-18, were not game mechanisms, but descriptors of real world human ability. In that way they were more like height and weight when written on a character sheet then any kind of numerical game mechanism.
By having a real world understanding about these six abilities, players could ostensibly predict how likely they were to succeed at a wide range of actions within the game world. The actual game mechanisms tied to these scores were not meant to be known to the players at all. Otherwise the players were judged to be playing a numbers game rather than a roleplaying game. By using these descriptors, real world knowledge could then be used to overcome challenges in the game world according to how well the players understood those challenges based upon their correspondence with reality. I should point out, that testing real world knowledge and ability was widely, but inaccurately considered a requirement of, if not the essence of roleplaying when the hobby began.
So fantasy, non-real world elements were defined within the game rules exclusively using what was believed to be real world descriptors in order that the players could understand and interact with these fantasy elements using each player's own real world knowledge. To the best of my research into this, the purpose behind all this tying in of game mechanics to reality was because roleplaying was understood differently than it is today. For at least 40-50 years roleplaying was thought to deal with reality, not fiction. Sadly, this "not a fiction" myth was held by both psychodramatists and role trainers. In RPS, or role training, roleplaying is still widely held by sociologists as unrelated to theatre. So, within this era's widespread view, by adding in fantasy, non-real world elements D&D was considered "not roleplaying" by most people when it came out. It's my opinion that most of this game design and the philosophy behind it was an attempt to justify each and every element in the game by using real world terms and, especially, by testing participant's real world knowledge and ability. All this because this testing was largely believed to be the primary objective of roleplaying.
So, fantasy races were termed "demi" humans, or semi-human "races" that could be played as long as their ability scores fell within the corresponding human/player understandable range of 3-18. This I'll call the PC-Playable range. One which was strictly limited to the 16 segment distribution based upon our own reality. Unlike humans however, fantasy races were understood to have their own demographic distributions specific to their population's characteristics. And when these racial distributions overlapped the PC-playable distribution, then certain classes were either fully playable, partially playable, or not roleplayable at all.
Here are some examples:
Halflings were weaker than humans. By my understanding, a PC halfling's Strength score corresponded to any other PC's Strength score on the 3-18 scale, but that halfling's score location upon their own race's demographic distribution was much higher. I don't remember the numbers exactly, but an 18 on the hypothetical "Halfling Racial Strength" table equated to the Halfling Maximum Strength score on the PC 3-18 table. Maybe an 8 IIRC.
Dwarfs could effectively be fighters to the highest limit of their race's distribution (8th?), could be thieves within the full spectrum of human understanding (unlimited), but were either too wise or too unwise (or perhaps just too inhuman) to be be playable as clerics (N/A).
All six abilities scores worked the same way defining each demi-human race according to the real world understanding of human abilities. Scores outside the 3-18 range were considered impossible to roleplay because they were outside the spectrum of understanding by the players in real world terms - reality being the only the spectrum within which one could roleplay.
EDIT:
Lastly, I'm not exactly sure if ability scores were the sole reason some classes were judged playable or limited in their playability. I believe other traits had an effect on these assumptions as well, but I don't as yet know how.
