I would imagine that someone wants to play a half-elf because they want to play something that has something that neither elves or humans have, which at the very least includes the idea of character that doesn't quite fit into human or elf society. That very idea is something not existent within the idea of human or elf alone, and is an example of how hybrid races add something not present in the parent races.
The idea of the half-elf draws on the concept of a more mystical person who is somewhat separate from normal human society due to their unusual heritage, yet not as foreign and unknown as elves themselves. In other words, a person with special abilities or knowledge that dwells outside of society, yet is still close enough to help in times of need. Aragorn from Lord of the Rings, a distant descendant of a half-elf, fits this thematic mold pretty squarely. It is the existence of archetypes like this one that explain why the half-elf race in particular out of all possible hybrids is popular and enduring in fantasy and D&D. In short, a half-elf fills a completely different story role than either a human or an elf.
Fair enough, and I get that sentiment. However, Aragorn (nor any of the other Dunedain) does not
have special abilities possessed by neither humans nor elves - he's just a human who lives for an exceptionally long period of time, essentially. That's much closer, in my mind, to my hybridized approach.
Aragorn isn't a great leader because he's part elven and mostly human. He's a great leader because of his personal characteristics. Any and all racial characteristics he has are inherited from his parent races, not anything particularly unique to him or the other descendants of Numenor.
The kind of thing you're getting at is frankly more akin to the idea of planetouched, who aren't half-breeds so much as they are humans or the descendents of humans who were altered by magic in some way (albeit, in many cases, through magic-enhanced breeding). The mythological Merlin, for example, is a prime example: he's not half-demon, half-human. He's a human who was born with extraordinary abilities because his mother was raped by an incubus.
Half-elves are much more mundane in my mind. There's nothing particularly "magical" about them. The same goes for half-orcs or half-dwarves, or whatever. They're just two different races getting together. Tieflings, aasimar, and genasi, on the other hand, fit more into the mold you're describing. At least in my opinion.