The technical part is, at several companies (including the one I work for).

A lot of companies now rely on tools such as Codility / Leetcode / HackerRank for technical screening, or their own in-house tests.

How would you determine if the person is a good fit culturally via an automated system?

have a human-to-human interview asking behavioral and background questions

and automate all the whiteboard/leetcode parts with boilerplate Q&A

Often times the most valuable signal comes from how the candidate got to an answer rather than the exact answer that the candidate got to.

So the entire purpose of the whiteboard interviews are to determine if the user got the answer, that's it? There's no other elements in play there?

this should be similar to GRE, SAT or any other standardized IQ test

How do you determine the difference between finding "a good fit culturally" and furthering a monoculture?

One approach is to define a set of values and communication habits that you believe you need people to be aligned on. That way, you allow for natural variation outside of that and you avoid giving your culture-fit interviewers a vague task.

How would you automate it?

So basically you're asking for a HackerRank screening. In my experience, if your code doesn't pass 100% of all test cases you get automatically rejected. At least with a human interviewer, they can evaluate your thought process even if the code itself isn't quite correct. Trust me you don't want to me interviewed only by robots.