It is a response to the following:
1. "Why have more than one class at all? Why not just fighters? Answer : The classes feel different and work differently. "
2. "I would say "why have Wizard Schools other than Orthodox", but I think there's more here. 'Why I like that as a referee' has one obvious answer: It makes my players happy/come to my games, and that makes me happy. I'd say it does add a small amount of extra complexity, but not needlessly. It's needful complexity. Why would you have only one casting class? Would that makes it more enjoyable for you to DM? Some other metric I'm not thinking of?"
Each class serves a distinct mechanical purpose within the OSR gameplay loop and adding, removing, or altering the playable classes may disrupt this. On of the strengths of OSR is the robustness of this loop in games and how it serves to produce the OSR playstyle. It is important to know how each class interacts with the OSR gameplay loop as it informs how other classes may be added.
OSR Gameplay Loop
The resources in the loop refer to both literal resources like light and rations as well as meta-resources like those outline on a character sheet like HP or Attributes. Resource drain is anything which may remove these resources such as traps, monsters, and time.
So aside from race as class, there are 4 default classes in the vast majority of OSR. They are Fighter, Magic-User, Cleric, and Thief. For this analysis we will ignore the thematic elements of each class and focus on their mechanical components.*
The Fighter class present consistent and predictable abilities (dealing more damage and taking damage from monsters at a reduced rate than other classes) to wager HP against the most common active HP drains, monsters.
The Magic-User class present a limited number of abilities (spells) with which to skip certain specific resource drains.
The Cleric class present limited resource renewal (healing hp, curing curses and disease, raising the dead) which serves to extend the amount wagering possible and serve against a specific type of HP drain the Undead, as such monsters lack morale and reaction rolls instead always being hostile and fighting to the death.
The Thief class presents consistent and predictable abilities versus the most common passive HP drains (finding or removing traps) which also serve to increase the rate of treasure gained (picking pockets and opening locks). Lastly thieves also have passive avoidance to active HP drain (hiding in shadows and listening at doors).*
*The goal of a thief is to combat traps and locks the same way the goal of a fighter is to combat monsters.
The other three common OSR classes: Elf, Dwarf, and Halfling can also be analyzed mechanically.
The Elf Class functions as a mixture of the Fighter and Magic-User classes, they possess nearly the same combat abilities as a Fighter with the exception of smaller HD and they cast spells as Magic-User could.
The Dwarf Class functions as a mixture of the Fighter and Thief classes, they possess nearly the same combat abilities as a Fighter with the exception of not being able to use bows or two-handed swords and possess the Thief's avoidance of traps.
The Halfling Class functions as a another mixture of the Fighter and Thief classes, instead having a thief's passive avoidance to active HP drain and nearly the same combat abilities as a Fighter with the exception of smaller HD and not being able to use bows or two-handed swords. However, they gain an extra bonus to their AC. Further Halflings have the best saves of any class, which means their chance of surviving random misfortune is greatly increased.
I would posit that the Elf and Dwarf class are redundant and don't add novel ways of interacting with the OSR gameplay loop. The Halfling class can be better described as having a reduced chance to have their resources drained and functions to allow greater wagering similar to a cleric. However, while a cleric returns lost resources a halfling instead has a lessened chance of losing resources wagered.
Thus we are left with a core 5 OSR classes:
Class One: Deals with Monsters
Class Two: Sidesteps wagers of resources for treasure* a limited amount of times
Class Three: Increases number of wagers of resources for treasure possible by resource return
Class Four: Deals with passive resource drains/increase rate of treasure acquisition
Class Five: Increase number of wagers possible by altering chance of it losing wagers of resources for treasure
*This refers to the standard mechanics of dungeoneering or hex-crawling. A spell like Sleep can sidestep combat and in doing so, ignores combat's standard wager of character hp vs enemy hp.
If one was to add further classes to an OSR game, one would have to have them interact with the OSR gameplay loop in another way or they would otherwise be redundant to the gameplay loop and thus to the game.
A quick aside for Racial Classes
Race as Class often subverts the gameplay loop in ways I didn't mention and has implications for the game-world as well as they merge the narrative and mechanics together. Further Racial Classes often function on different mechanics than the core 5 classes, such as not having HP, being able to see in the dark, or being able to breathe flame. Further, racial classes should have a distinct feel that's different from the human classes to emphasize their inhumanness.
So, in regards as to "Why would you have only one casting class" I would rephrase it first as "Why would you have only one class which sidesteps wagers of resources for treasure a limited amount of times". I would say that if there were two classes which "cast spells" it would only matter if they both interacted with the OSR gameplay loop in the same way. In fact B/X has two casting classes the Cleric and the Magic-User but they interact with the OSR gameplay loop differently.
In the time I have been writing this +Arnold K. has brought up the examples of two different casters
"(a caster that loses Con when they cast spells will feel different from a wizard who spends gold coins to cast spells)."
I think these are both distinct enough to be offered in play as they interact with the OSR gameplay loop in a novel way. The class which loses a non-renewable resource to sidestep wagers of resources for treasures is different from a class which loses treasure in order to sidestep wagers of resources for treasures is different from the standard class which uses vancian preparation to sidestep further wagers of resources for treasures.
However, if you had two class which both interact with the OSR gameplay loop in the same way or in excess similarity why not consolidate them into a single class? Why have a warrior, knight, and barbarian class instead of having a single fighting-man class. If each just deal consistent damage, have lots of hp, and may wear armor why give them minor differently mechanics to interact with the OSR gameplay loop.