I agree. Dog theft is a serious problem now and unless I know you really well and trust you 120%, it's just not going to happen unless you're a legit business with positive referrals and insured and bonded. I used to walk and play with the neighbors dog(s) as a kid but dog theft wasn't a problem and the owners knew my parents really well. Mind you, I'm 27 so I'm not as old and that last sentence made me sound, but still.
Those who do allow you to walk their dogs currently, why don't you ask them if they'd be willing to write letters of recommendation for you to keep on hand. Also, tip, never forget to carry business cards with you. If I had been in his position, I'd have said no as well because it's a little fishy.
I have a lot of positive referrals who really trust mr with their dogs. Hence, we go on hikes just me and the dogs independently.