A lot of people are arguing from either a mechanical or social justice perspective. I would posit that these are, if not exactly wrong, missing a certain fundamental point. Namely, races are about character and story.
Personally, I don't like nitpicky character building in my D&D. I don't mind it in some places (eg: Mutants and Masterminds), but for a fantasy RPG, it really grates. I don't like Pathfinder, I didn't like 3E/4E, and AD&D's splatbook hell chased me away from D&D for a long time. 5E's simplicity and streamlining brought me back. Level Up manages the careful feat of giving more flexibility in its character building and leveling up without getting too complicated.
Obviously this is entirely anecdotal and of limited sample size, but in my experience, most players (2/3? 3/4?) do not want to play Fantasy Spreadsheet. And similarly for a character's backstory. Players may develop a character's backstory more over time, but more than a paragraph or two to start with is already a big ask. D&D addresses that by giving package deals on backstory tropes/stereotypes/characterizations. These are called "races".
Note that D&D races are absolutely not the same thing as real life races, nor are they species. They are ideas given shape, and players can choose one in order to match a general idea of what type of character and story they want to play. They can lean into type (a dwarf cleric) or against type (a halfling barbarian) based on how they want to play with the available tropes, to fit the character into the idea of what they want to play. The mechanics, of course, help support those ideas, but it's not the mechanics which define the ideas.
Anyway, these two points come together in what I feel is one of the biggest draws for D&D's current popularity: the startup costs for both mechanics and backstory are very low. Increasing the cost of either is a decision a game designer needs to consider very carefully. For example, a point-buy race builder is a cost not only for the designer to create and balance, but for every player who not only has to spend extra effort creating such a race, but who has now been deprived of the generalized concept to build the character's identity on.
If an existing race is not being used, or its story tropes do not lend themselves well to a decent chunk of ideas (even if the race is "good" in a mechanical sense), then it may be worthwhile to drop the race in favor of something else. However if a race is being used, and it fits character concepts that players want to explore, then it absolutely should stay. The designer's job is to create the board for players to play on, not to tell them they're having bad-wrong-fun.
In the case of the half-elf and the half-orc, and the general outcast idea, I have my own impressions of them. Both are representing outcasts, but of two different types.
The half-elf is the unwanted outcast that withdraws into himself (ie: emo outcast). The unwanted child, the noble dalliance, mingling between the Montague and Capulet families, or the general disdain the "nerd" may feel among the popular kids. The individual outcast.
The half-orc is the downtrodden underclass outcast that fights back violently against society. The former slave, the Irish immigrant, the kid from the poor part of town. The societal outcast.
Note that a half-elf doesn't have to be (or even be likely to be) an emo outcast, but if you were picking out emo outcast characters, I'd bet the largest percentage of them would be half-elves. Just try to think of an emo outcast dwarf, or elf, or halfling, or even half-orc. It just mostly doesn't fit. I'm sure you could create such a character, but you're taking on the work of doing that yourself.
The point of the "race" is to get you most of the way to where you want to be with minimal work. That's its purpose as a function of the game's design. It's a system that's shown to work quite well in D&D. At the same time, each race has to give you a fairly broad spectrum of options. A half-elf isn't only an emo outcast, for example. If a race can't provide a decently broad set of easy characterizations (even with some overlapping other races), it also fails in its purpose.
There's another part of the "outcast" concept that ties directly to how an RPG works: As the characters grow, they overcome their former weaknesses. The outcast, the underdog, proves himself to be worthy of respect, of being capable, and even heroic. Whether that's Rocky or Naruto, these types of stories have existed for a long time. When you choose to start with an underdog character, there's sort of the implied assumption that your character will eventually be able to show the world he has true value in the end. People love underdog stories. They want to see people overcome adversity.
When you take away those types of races, you're taking away some of the easy ways for people to know that they can explore that kind of story. Again, you can do it manually, but it's more work, and many players won't even recognize that they have that as an option if it's not readily available.
So, attempts to frame this as purely a mechanical issue, or solely about real-life prejudice, I feel is fundamentally detrimental to the purpose of their existence in the game system in the first place.
As an aside: Level Up's approach shifts how this is built. Most of what I've described above is more part of the selection of Culture (including things like Lone Wanderer and Tyrannized), so Level Up can handle mixed races at the more mechanical/biological level of Heritage. Looking to them for the fix for how to create half-races solely at the mechanical level misses that a major part of the issue is addressed in an entirely different way in that system.
