See, I don't get half-elves.
"*I* would not like to play them, and therefore *YOU* shouldn't play them," is what you are saying.
Here is a thing that all GMs should learn... Most of it isn't about you. You don't have to get it. If *they* get it and have fun with it, that is what matters.
Yes, a GM should have fun, and the GM is certainly steward of what is available in a campaign. But, that power should be used for the right reasons. Are the half-elf or half-orc mechanically broken, so that they'd hose the fun of other players or the balance of your adventure design? That's unlikely. From there... does them playing something you don't personally find fun really bother you? Should it?
I, myself, don't like coffee. To me, it tastes of ashes and sadness, and I just don't see the appeal. However, I know a lot of other people do like coffee, so when I have folks over for dinner, sure, I set most of the menu, but I can and will produce coffee to go with dessert. Because that coffee is for them, not me. My pleasure is not from the coffee, it is in giving them a meal they enjoy.
Now, If I were allergic to coffee, or the smell of it gave me migraines, I could see saying that I won't have it at my table. But just for not liking it myself? That's not a great reason to say someone else can't have it.
Half-elves and half-orcs are described often as anomalies or one-offs. That they're all searching for their place in society, or their parents don't accept them, or blah blah blah. Yeah, it worked for Dragonlance back in the 80s, but that was thirty years ago. Now it's not only cliche, but boring. Not to mention, half-orcs are portrayed a lot of the time as products of horrible assaults.
Aside form the fact that thalf-elves had been in the game for a decade or so before Dragonlance, and in Tolkien for decades before that... Being alienated is a standard problem of real-world humans. It is a major element in young adult fiction for a reason - many people identify with it.
Yes, Half-orcs are often portrayed as the products of horrible assaults. But then again... how many sentient creatures do adventurers kill in the course of their career? If horrible assault was really your issue, a game that a lot of the time is a bunch of murderhobos killing things and taking their stuff may not be your best bet. In playing D&D, you are generally accepting that you are playing a world of violence, and will either have to gloss over the impact of that, or take the effort to address it. If you are glossing, then the half-orc isn't an issue, and if you are addressing, the half-orc can be a useful thematic agent.