Personally, I think with very strict booking rules I don't need to ask for a deposit. I get people offering deposits to get round my screening methods

That makes me feel uncomfortable; offering monetary incentives to get around my personal rules... it's a no from me.
I think there's no way to please every SS. Some don't like my screening, some don't like deposits. If a provider won't see you without something you're not willing to do, then shrug it off and find someone who will.