According to classic D&D lore, there is no "surviving" becoming an illithid; if the tadpole fulfills its biological purpose, it takes over the host's form completely and irrevocably. A "half-illithid" comes about when a tadpole is inserted into the brain of a non-human. They are not as powerful as true illithids, but are much stronger than normal specimens of their race. My point in all that is that a "half-illithid" isn't going to be something that could work in this instance, at least by RAW canon.
When you say "classic D&D lore," do you mean pre-5e? If so, I would think it fair game to change what a half-illithid means in 5e home-brew, especially if the term "illithoid" is used. The description in Volo's (p. 72) uses "humanoid" instead of "human" as a host with no mention of "half-illithid" that I can see.
As for surviving, I would say if the process is interrupted or if the tadpole is not strong enough to eat the entire brain there may be a chance that an illithoid is formed.
It IS possible for an illithid to sever ties with and flee its community and Elder Brain, though this is normally done in the pursuit of magical knowledge and eventual lichdom. So that could work as a background if it fits what your player wants to do.
I would allow but not require that.
Realistically, I think your race writeup will work fine. The biggest hurdle the PC will have to overcome, as others have noted, will be the fact that they will terrify almost everyone they meet, who will likely resort to violence immediately. It's not wise to take chances with mind flayers. That said, there is precedent for illithgids to trade with and have strong connections with other races, from fellow Underdark denizens to basically everyone in the Astromundi Cluster. I guess if you want to play it that way, there's no reason ind flayers couldn't be an accepted (if distasteful) trading partner to a kingdom. As their reproductive cycle is a closely guarded secret, even that might not deter greedy merchants from dealing with the illithids. And that's assuming the creatures are well-known enough to even be feared and distrusted beyond their obviously alien appearance.
I think it just depends on the campaign world.
If mind flayers are known to surface worlders, then the illithid would likely need to hide itself under cloaks or with disguise/alter self spells.
If mind flayer are only known in the under dark, these PCs would be strangers, grotesque, but not hated.
Though either way, they might be hunted by other illithid and gith alike.
I think some other flavor ideas like "Food for Thought" (Volo's p. 71) that says they acquire stray, fleeting memories from brains they eat and cultural sophistication such as feeding on gridlock brains makes them pay almost no attention to crafting/styling clothing.
I'll also point out that Illithoids could use the Personality Traits, Ideals, Bonds, and Flaws on pp. 74-75 of Volo's.
FYI: I'm making changes to it at
http://homebrewery.naturalcrit.com/share/ByQwUAh2eb (also found on the first post of this thread).
