The "default" flavor of core can be simple:
• Magic exists
• Culture is vaguely medievalesque
• Humans only!
• A note can mention the DM might add other races.
To select nonhuman races for a particular setting is an important way to create the tropes, themes, and tone of the specific setting.
I feel each setting should present its own assemblage of races.
Of course, a DM can plug in a race from one setting into an other setting.
Make the Forgotten Realms Guide a separate core book. It is an official setting that details the Forgotten Realms setting assumptions including specific races, cultures, religions, and planes. The Player Handbook doesnt go into these kinds of setting assumptions.
The Players Handbook is a rule book with all of the rules that are necessary to play the game, including combat and skills. Pretty much every setting will need to use the Players Handbook. But not every setting with use the content in the Forgotten Realms Guide.