All three metrics you mentioned can be evaluated using writing. In fact, writing makes up most of the communication in a remote environment.

I will think in this direction, thank you for putting some more arguments towards that!

My main concern with this direction is whether we should evaluate for language speaking skills. Second concern is cheating. Third concern, it's more difficult to build.

It's actually easier. Cheating can be discouraged via real-time chat sessions, while async communication like emails or essay-style discussions can be scored via an NLP library like spaCy. If a machine can understand and parse the candidate's writing, chances are it's valid English.

That's an awesome idea! Thank you! I'll talk with some users about it to hear their feedback.