Unfortunately there is always a possibility that discriminatory attitudes will influence hiring at any stage of the process.The partner interview is discrimination and the best way to get an offer is to have a partner like you.
It becomes a lottery based on who your interviewer is and how they perceive you. There is nothing fair or objective about it.
For example, the name on your application will give away whether you are male or female and may also indicate your race. Even if your name is hidden when your application is reviewed, the type of work experience or extracurriculars you have can, for example, indicate your class background. And if we zoom out even further, structural disadvantage makes it harder for people from certain groups to even be in a position to make a competitive application to begin with.
Of course, at an interview, it's easier to spot people's race, sex, disability etc. and take a biased view, but this would be a problem even firms they didn't allow partners or associates to interview you, as you suggested, because any interviewer will have some level of bias. I'm sure every firm evaluates its processes to ensure they are as objective as possible, for example through competency or commercial interviews where candidates are given numerical scores based on the content of their responses.
At the end of the day, though, law is fundamentally a people business, and firms want to hire people who work well with others and can get on with clients. Sometimes someone gets a bad interviewer who makes it impossible to build a rapport with them, and that's the interviewer's fault. But if someone is consistently failing to build a rapport with interviewers, this suggests a need to brush up on people skills. Given that most people can develop people skills with practice, and that these skills are essential to the career, I would argue it's not discrimination to reject someone who lacks such skills.