I know the common wisdom that every champion is viable in any position (for the most part) so long as you know what you're doing, but I think I missed the reason Kha'Zix fell out of popularity in solo lanes.
he isnt very popular in LCS right now because reasons. and since noobs always copy LCS and 99% of the community are noobs, there you go.
khazix was a solo laner when hes poke was hes burst (think zed or talon even if they need to be melee for a proper burst they max ranged spells first) riot though nah nah nah khazix damage is hes q when evoled so they took a shit ton of damage from w and threw it on q this pretty much killed mid khazix but due to this buff to hes 2 second cd q top khazix became a thing and then riot was like... welp... hes one shoting the island lane... we need to sort this out then he went threw a blender of nerfs chances and buff and we get this. a alright khazix who suited well for being a jungle assassin only problem is jungle assassins arnt really in vs this tank meta but vs stuff like nidalee jungle khazix will easily win that match up with hes isolation and just flat out faster burst. hope this explains alot
I believe the reason he was removed from lanes was due to the changes to his Q. Not only it is far more difficult to isolate a target, you no longer get execute damage for using your evolved Q.
He used to use his Q and it's execute damage to steadily chunk down his lane opponents, but now he doesn't get any bonus damage if they are near their minions which hinders his trading potential.
Assassins have fallen out of favor because they do not do well against tanks (we are in a pretty tank heavy meta), and kha'zix in general is not played often anymore (especially in a solo lane) because the isolation range on his Q (which is most of his damage) was greatly increased making it much harder to isolate the enemy champion.
The main reason why he sucks in lane is the same as 
and the likes, he has too many losing matchups and incredibly feast or famine, meaning if he gets a kill he instnatly goes 10-0 , if he dies early on 0-10 is plausible. And he has no fallback pattern since as laner you cant just abandon ur lane and go into the jungle.
because his only way of sustaining the lane is also his only way of csing-harasing from distance , even mele meta midlaners/tops like Diana and rumple have enough range harras to punish kha zix each time he tries to come in for a cs .
