Archmage technically doesn't progress caster level automatically by RAW. All you get is spells per day and spells known. You have to spend High Arcana on that. Arcane trickster only gets spells per day. Eldritch knight only gets spells per day and caster level, but no spells known. Loremaster gets all 3. Mystic theurge only gets spells per day and caster level, it doesn't learn new spells. Red wizard is spells per day and caster level only. Thaumaturgist is spells per day and caster level.
As you can see, loremaster is the only one that fully progresses the base class. PRCs were intended to be altered for your game/setting. Saying they all get full casting progression doesn't hurt anything really, and future PRC templates relaxed the specificity of how spell progression worked anyway. Technically, if Archmage gets caster levels without spending high arcana then hierophant can absolutely progress spellcasting.