These guys bring up a few good points. Namely, how T'Challa could have pointed out that Killmonger broke Klaw out of custody and help him slip from being held accountable for his crimes.
He swore to bring back Klaue and failed.
Trying to make excuses for his failure ("But it wasn't my fault, Killmonger outsmarted me!") would have just made him look even weaker than he already did coming home empty-handed.
Having a strong king is very important in Wakandan culture, hence the ritual battle used to determine who sits on the throne.
Hearing that Killmonger got the better of T'Challa before he showed up with Klaue's dead body would have just made W'Kabi even more disgusted with T'Challa.
And hearing about it
after Killmonger showed up with Klaue's dead body would have just made W'Kabi even more determined to support Killmonger: The stronger man, the man who was able to get the job done.
Also, the fact that Killmonger's father working with Klaw makes his father a traitor, which would seem to nullify his right to challenge T'Challa.
There's no law that says you're ineligible for the throne based on your father's crimes. All you need is royal blood.