Would it be reasonable to have an invitation from a person with x reputation bump someone up to the required rep to ask questions? One could spend time in meta, and the chat room where people could get to know them. For the well-intended but naive, this would encourage getting to know the site and expectations of the culture. For the people who are just a bad fit, they could be filtered with minimal effort. Would there be any undesirable consequences from this approach?

Who do you imagine would be the user that would bother going through this interview process? If you've used the site before you already know how strict the moderation is on questions keeping people from asking questions already, do you really want to add a process where the user has to kiss up to the moderators for a couple weeks before asking their question as well?

It just would not be worth any new users time, especially given that when a user wants a question answered, it tends be slightly time sensitive.

I've given this some consideration, but they have put the barrier to entry to the chat feature above a minimal reputation ranking.

One can easily overcome that with ten skillful edits, 2 upvotes (or 1 upvote and an accept) on an answer, or 4 upvotes on a question (20 rep is required for chat).

It's fairly easy for an individual to get into the chat, if they want.

That barrier does keep spammers from flooding the chat, though. And if you consider it an interview process, that's what the earning of reputation points accomplishes, and I'd argue it does so rather elegantly.