In my last campaign, I had a reduced set of available races, with the majority of the intelligent people being Human. The ancient, all-but-nearly-forgotten, past once was very Forgotten-Realms-like in it's racial variety. However, there had been a terrible war that involved ALL races on one side or the other. Kind of like a super-sized LOTR style conflict that raged for a hundred years or more. The ancient races of elves, pixies, fairies, dwarves, gnomes, etc. were no more and commonly believed to have been pure fiction. This belief was greatly aided by the average human commoners' knowledge of those elder races having become corrupted and simplified over time...
Pixies, fairies, brownies, and elves had all become combined in the cultural memory as a single fey creature called Elves proper with some of them being given other names for supposedly the same creature. They were believed to be pointy-eared humanoids about 8 to 12 inches tall, very slight of build with colourful butterfly wings. Big fairies basically. They were magical and could easily cast all sorts of magic, mostly of an illusionary or glamour like nature. The "reality" of it was that the historical separate fey races had been as described in the PHB/MM. Those that weren't wiped out in the conflict, had all simply died off over time. Long life spans and slow breeding cycles don't exactly help one out in a violent world. The Elves survived, just barely, and decided to take off for parts unknown and avoid all the other races.
Dwarves, Gnomes and Halflings were similarly combined into a fictional creature that was physically the equivalent of a 1-2 foot tall garden gnome and were believed to live in the forest or under the ground doing secretive things both dark and slightly magical in an "imbuing magic into items" kind of way. What had actually happened in this case was that the gnomes were literally exterminated in the ancient war, the halflings were magically corrupted into sub-sentient clawed monsters (think baboons with 8 inch curved claws on their fingers), and the Dwarves were driven FAR underground in a losing battle against an army of super-regenerating undead. The Dwarves eventually destroyed the undead, but also depleted their population below a sustainable level in the process. Most Dwarves stoicly accepted this and became extinct. A small few decided to embrace a dark magic that forcibly blended their bloodlines with humans in a limited way. This Dwarven offshoot are currently living amongst humans as 5 foot tall, stocky and muscular "humans". They can interbreed, but the result is always one of these "blended Dwarfs". Their con and str is Dwarven, however they only live to about 175-200 max. (this is something that they do their best to conceal from the (fellow humans") In return for what they've lost, they also can breed at a human rate. They have no other Dwarven or Human racial traits.
The other surviving sentient races were small packs of Goblins in the mountains (they now only bred like humans instead of the traditional accelerated "rabbit-like" breeding commonly used, although they don't live any longer than before.) Orcs existed, but only in a single region. They weren't as strong as they once were, but they also weren't as dumb either. They were slowly interbreeding themselves out of existance by combining with humans again and again over the generations. There were several races of lizardmen, but they were only really comfortable in wet humid swampy regions and thus kept pretty much to themselves in an empire that was considered pretty creepy due to the super polite and oily mannered diplomats occasionally sent to Human territories. They would never reveal all they knew and usually nothing about their facist and zenophobic motherland.
Ther rest of the world was Human. I made several human cultures. A "City-state European" one, a "musical Chinese" one, and a vaguely "middle-eastern/near orient" culture. Each culture had their own advantages and disadvantages.
All my players had Human characters from one culture or the other and they all had a blast. No one felt that the campaign was "lacking" due to the absence of the other races. Why not? Not because a campaign "doesn't need" the other races. The reason it was a success was because it was a good campaign. High quality is high quality no matter what specific pieces it's built from. It's all a matter of personal taste and style being in tune to the quality of the individual game sessions and also matching the overal campaign theme and style.
I've run campaigns in settings with lots of different PC races, both traditional and non-traditional. They were greatly enjoyed as well. I've also run games that were really bad too, and when they stank it wasn't due to the presence or absence of any specific races. Just as high quality doesn't require a magic list of specific ingredients to be good, no amount of additional ingredients will fix something that is crap. Garbage is garbage no matter what it's made from.
