When dogs seem to want to make contact, but are skittish, it has worked well in my experience to get them comfortable on a leash as soon as possible.
Allowing them the freedom to flee actually prolongs the socialization period and can make things worse. They longer they can escape your touch, the more anxious they become.
There are several reasons a seemingly friendly dog becomes skittish when approached, leashed, or touched. But the main reason is fear because they’ve had bad experiences being caught, leashed, or touched.
In the video, Meagan uses treats to lure the shy dog into a looped leash, she eventually left the leash drooped over him and removed it when he returned for a treat. Once he was okay with the leash she let him lead the way in the pen and offered treats to get him moving rather than pull.
When he became comfortable being leashed and walked in a neutral area, it was time to take a walk. She let him sniff, let him lead, roll and gave treats occasionally. Meanwhile, she would stroke his back once and a while, but nothing intrusive or constant.
Dogs like to discover new smells, so their nose will lead the way. For extremely shy animals a trail of treats and or a more confident companion dog is very useful in distracting them from their fear.
Walks help build a bond of trust, as long as you take your time and make it a good experience. If you only leash your dog to get things done or take them to the vet it’s counterproductive.
In the end they will come to trust you, enjoy the walks and come to you for affection.
UPDATE: Malcom, the dog in the film has been adopted and is doing great in his new home.