Ahhh, the old spirit of hand-rubbing-phone-receiver lives on: “What? Say again? Sorry, you’re breaking up.” 
I suppose I might say something like “what would you like to know beyond what’s in my profile and application?” I mean, like you said, hopefully the relevant information is already in the profile, and I’ll have said in my application why I’m interested in the sit. Turn it around to them, what else specifically do you want to know. Maybe they don’t even have an actual question.
As to the “job interview question”, I think whether that would bother me would depend on the tone with which it was asked. Maybe the HO is used to interviewing job applicants, maybe it’s just their manner, and they all want to make sure they have someone competent and reliable so they’re looking for an example of why you are that. 
But then I’m generally not bothered by the “job interview” idea anyway – that’s just me.
I think this is a nice way to do it if anyone’s uncomfortable outright saying “I don’t think we’re a good match” or “sorry, I don’t do…” in whatever variations.
For the OP/sitters, this could be flipped to “I’ve got another chat scheduled for a sit that would overlap this one; I’ll let you know” and then you could message that you accepted the other sit.