Most of those classes didn't exist until well after the game shifted focus away from wargaming though. You're right about the Fighter (originally Fighting Man) just being the basic soldier. Wizards might have been thought of as artillery, or at least the Fireball spell was. But those were the first two classes and that's where it seems to stop.

Clerics were definitely patterned after Catholic priests on medieval battlefields (heavy "knight" armor but only blunt weapons) but their abilities were oriented toward aiding one or two elite combatants, not patching up a field hospital full of wounded infantry. The fourth class added to the game, Thief, was pretty bad in a fight and got all of its value from interacting with dungeon features like traps and secret doors. (Old school thieves sucked at THAT too because of some weird design decisions, but it was the focus of most of their rules text and clearly what they were "supposed" to do.)

More content was added later through magazine articles and house rules at local gaming clubs and TSR's internal work, after the game had thoroughly shifted from "wargame you can also use for dungeon crawls" to "dungeon crawling game where you might do some mass combat if you got go high level". You mentioned Barbarian as heavy battlefield infantry, but the AD&D 1st edition barbarian was a reworked fighter designed for taking point in dungeons. He gave up heavy armor which in 20th century D&D was most valuable against low level mooks like enemy footsoldiers. In trade he got bigger single attacks (good against big fantasy monsters, but mooks would have died anyway), more hitpoints, and most notably bonuses against surprise and traps.