Nick,
You're absolutely right. It's confusing as can be right now for everyone, even us.
Until a few months ago, I would have (and have plenty of times in these forums) said you don't need to go through the review process. If you're a Facebook admin, you can get any permissions you need, always, without going through the app review. App review was only if want to get permissions from your general users. Easy peasy.
However, Facebook has been less than clear with all of the recent changes as to what will happen on August 1st. In some places, like you mention, they show that app review isn't necessary if you're just getting permissions for admins. Other places, they state that app review is required for all permissions, whether you as an admin are using them or everyone. Some permissions have even been restricted to need app review before you can even test them.. which is pretty ridiculous since you need to show how you use them before you can go through app review.. so it's a chicken and egg with those few.
Our recommendation now is to submit whatever you can as a precaution. If you're ok with your stream possibly not showing after August 1st, you can not try to go through the review process now and see what happens. We'll agree that going through the app review process with how you're using the app right now would be difficult, but if you want to at least explain how you're using it, it can't hurt.
After August 1st, we'll know more based on what happens. It's very opaque right now though and Facebook's documentation is all over the place as they're moving very quickly to implement a lot of updates.
Sorry I can't give you a definitive 'do this', but hopefully the above explains the situation better.
Thanks,
Alex