OP I think you were NOT being unreasonable.
I do, however, think you need some kind of assertive "Please don't bring your dog any closer" line ready.
I go to the playpark at the end of our road a lot, it's a swingset and climbing frame and slide in an open field, that backs onto a playing field and then a long trail through the local woods. Because of this there are regularly dogs around, and I often walk my dog and toddler through the woods and field before we end up at the swings (bribery in action!).
Sometimes my dog has been running off-leash through the woods and fields (allowed, appropriate) and if there are other children at the park, I then leash her. If there aren't, I tend to let her roam around. When families arrive, I leash her and say something along the lines of "I don't know how your children feel about dogs, so I'm going to tie her here. She's calm and very used to babies and toddlers but she might bark at strange men, specially wearing hats. If they want to pet her I will come over and introduce them."
The majority of families in this area say "oh, let her run, we have a dog/they love dogs/can they come and stroke her?" but a few are slightly hesitant or even just say "they're not so keen" and we always move right away, at once. I cannot see how shoving your dog at a child who's afraid of dogs will do anything but upset them! She was being unreasonable.
What I'm saying is, as a dog owner who also has a child, I'm happy to be told "please can you move your dog?" at any point, and I try to make sure no one is upset. It really helps if people say out loud and clearly "they don't like dogs, please keep your dog away" - anyone who ignores such a statement is being extremely U.