Or can they?
Men & Magic, p. 8:
"There is no reason that players cannot be allowed to play as virtually anything..."p. 9:
"In order for men to change class they must have a score of 16 or better in the prime requisite (see below) of the class they wish to change to, and this score must be unmodified."AD&D limits dual-classing to humans only, but then AD&D also is less lenient about playing as "virtually anything", per that first rule. So why not require Dwarves, Halflings, and Elves start with the classes as described, because that is what their culture teaches them, but allow them to change class later if they have the 16+ score in the prime requisite for that class? The only restriction would be the same as for humans: once you pick Magic-User or Cleric, you can't choose another spell-casting class.
You might even allow surpassing the racial level limits for those with unmodified scores of 16+, if you're feeling lenient. AD&D did this, to a limited extent.