I want you to step in the way back machine and think about all the Dungeons & Dragons games you've participate in. Did it really matter what fantasy race a player chose for his or her character? Obviously there were mechanical reasons to chose one race over another for attribute bonuses, special abilities, and access to kits or prestige classes but what difference did it make in the campaign? Would there be a noticeable difference in the campaign had your human Fighter been a goliath, an elf, or a dragonborn?
100% yes, because I make it matter. In fact, I consider it one of my primary jobs as the DM to take the story hooks that the players have given me for their characters and figure out how to use them. If a player is playing an gnome in my game, then gnomes are somehow going to be in the foreground of my game. If nobody is playing an gnome, then it doesn't matter and gnomes will regress into the background and maybe never show up at all because some other collection of people are going to be taking center stage.
Having said that - nobody in any of my current games is playing a gnome, and honestly nobody in the last 20 years I've DM'd has expressed an interest in playing a gnome, so gnomes don't have a real role in any of the three games I'm running and only ever mattered in one game that I ran decades ago that used some vague stuff about gnomes from the Basic Set as a springboard for a few adventures. OTOH, one of my games has a dwarf in it so the dwarf kingdom is very important in that game, while my other two groups that have no dwarves haven't had a single reason to interact with dwarves at all and have been going down different paths. All of my groups have elves in them, and honestly every game I've run ever has had at least one elf or half-elf in it, so in just about every campaign I've ever run the elves have some kind of recurring influence that I've had to think out (and in two out of the three current games, elven dynastic politics is a major plot point because they both contain players who are elven nobility of some kind - kids love their elven princes and princesses, I guess). One group has a dragonborn in it and another has a warforged (or "forgeborn" really, since it's the 13A game), and so dragonborn are "important" in the former but not in the latter and warforged are the opposite.
If you've got a setting where demons don't mate with demi-humans then tieflings probably won't work. At least not as written and then why bother having tieflings in the first place?
As an aside - tieflings do not have to be half-demons. When I've had them in the game, we've played them as the descendants of some other group whose ancestors were "blessed/cursed" with demonic power (which IIRC was the 4e explanation of them). No actual demon in their heritage, just a visible mark of the foul things their ancestors did.
Similarly, "half-orcs" in any game I runs never have any actual orc in their heritage - because orcs in my settings are never naturalistic creatures that breed with each other let alone with humans. They are always supernatural infections of some sort that reproduce through some suitably horrific means (which varies from campaign to campaign where orcs matter, but often either something like how the aliens in Alien do it, or a spawning pit if I want something less nightmare inducing). I use this background for orcs in all of my games - even if I'm running them in a published campaign setting - partly because I like to have something to distinguish hobgoblins and orcs from each other, but mostly because I like there to be some things that my players don't have to have any moral qualms about slaughtering (having been scarred by old-school modules where you enter a cave with 50 orc women and children, I suppose). When I had a player who wanted to play a "civilized goblin" we used the half-orc and he was a hobgoblin with a unique background, because if I have a player who wants to try out something for its mechanical benefits but isn't interested in the story aspects of it, or if the story aspects conflict with our established lore for a game, we work together to reskin it to something that both of us will enjoy having in the game.
I just don't think having a plethora of available races necessarily adds much of anything to the setting. Anyone with me or am I out on a limb here?
My attitude is always "see what kind of world my players want to play in, then shape the world around that". But I have a strong preference for gonzo worlds where anything goes, so I know other folks' mileage will vary.