As most of those bullets point out, nearly all of these are good indicators for some people and bad for others based on what kind of person they are or what their environment might be. It seems like the ideal would be to allow candidates to choose what format they think would best showcase how they work? Choose-your-own-adventure style?

Assuming you had some way of making comparisons across these different formats (easier said than done, I'm sure) it seems like you would end up with "truer" signals. It also seems like you would get an extra signal based on what the interviewee chooses. For example someone who prefers live coding/pair programming sessions might be someone who leans on and thrives more in ultra-collaborative/brainstorm-ish settings. Whereas someone who prefers take home assignments maybe is more productive being able to sit on a problem and iterate, then collaborating after being able to articulate the specifics in a code review type setting. Neither are wrong, nor are they mutually exclusive within an organization but in terms of team placement or general "culture fit" (whatever that means) it seems like something that's useful to know ahead of time.

Of course you might also allow people to choose which route they think they can hack more but I think that problem is more of an implementation detail of any given method. Asking for a solution to find the perimeter of a group of pixels at a video streaming company is better than asking someone to invert a binary tree at virtually any company. Asking someone to build something real-ish with a hundred valid solutions is better than asking someone to implement a trie with a thin search interface on top. And so on.