One of the biggest complaints I hear about B/X D&D is how race equals class for non-human PCs. The fact that race is class for demi-humans has never really been an issue to me for a few reasons:
1. In play it turns out to be identical to the earliest versions of D&D. Using the 3 Little Brown Books (LBBs), a dwarf could only be a Fighting Man so in effect Dwarf=Fighter.
2. If I met with real resistance or someone REALLY wanted to play a halfling thief it is not like it is terribly difficult to make up the combo.
3. Earlier versions of D&D (including B/X) were human-centric. There weren't suppose to be many elves or dwarfs running around exploring ancient tombs.
But likely the main reason for me not minding all dwarfs are basically fighters or all elves are a unique blend of fighters and magic-users was my earliest interpretations of these races. I began playing B/X before I read Tolkien, Moorcock or Howard. My baseline for what elves or dwarfs were from Moldvay's Basic rulebook and encyclopedias. My understanding of what an elf was therefore was basically the norse mythical version fit into the rule's description. One of the interesting things that first struck me and has always remained in the back of my head through later D&D description of elves and dwarfs is that they were related - something that I have often incorporated into my homebrew settings.
Often in my homebrew campaigns elves are a magical race of minor nature and fertility spirits, who are often pictured as youthful-seeming men and women of great grace and delicate features living in forests and underground places and caves, or in wells and springs. Being semi-divine they have no deities themselves. As such all elves are the classic D&D combo Fighter/Magic-user. Another interesting thing I sometimes incorporate is the Norse idea that famous men could be elevated to the rank of elves after death. Kill a player's famous fighter and let the party encounter an elf that is the elevated spirit of the former PC gives a real connection.
Dwarfs are the non-magic using brethren of elves (the mythological Dark Elves - this was before the Drow are mentioned in the first Monster Manual). I often make the creation of dwarfs the product of some cursed elves or some catastrophe. These non-magic using elves are outsiders to typical elves and either through self-exile or enslavement end up underground (think Morlocks). Therefore, as non-magic using cousins to elves, dwarfs cannot be magic-users and do not have deities so cannot be clerics.
Halflings are a bit different in that there isn't the same depth of mythology about them as there is for elves and dwarfs. I really just relied on the description in the rulebook but since elves and dwarfs were their own classes I never questioned it for halflings.
Only humans have the lust and greed to be thieves.