In a way the vet is probably correct, since it's a virus it just has to run its course.
Is your broody hen somewhat separated from the others? There is a good chance she may not even get it and transmit it to the chicks. My hen (long story) refused to be moved from the most popular nest box, and (again, long story) set and hatched right in a central flock location. She also became pretty ill - lesions and probably wet pox - just a day or two after her chicks hatched. I think the timing and situation created the perfect storm with the new chicks. She hatched two and one died in an unrelated freak accident. (Note: we do not normally have this kind of reckless trauma with broody hens. This one had several tries and fails that summer.) The second chicks contracted pox around one week old. He had terrible lesions, ended up with severely stunted growth, and seemed to have some type of brain damage. He just ran around frantically peeping - always. We ended up culling him after about a month when it was evident he wasn't going to pull through.
Our other chicks were all older when pox went through. Maybe 4-6 weeks, but still with the hen. That hen contracted a milder case and one chick also came down. That chick had pretty severe lesions, but still functioned well and had no lasting effects.
My understanding is that all of my chickens that had pox are now forever immune. New ones could come down with pox if exposed, but the immune ones will not. So, the vaccine would protect any new chickens. Although, what I'm reading looks like you vaccinate chicks when they're older. I'm not sure about it staying in the soil - I thought it was spread through mosquitoes, actually.
If I were you, and you aren't afraid to cull, I'd let your hen go ahead and hatch her chicks and see how it goes. By nature, hens usually keep their chicks away from close contact with the flock anyway, although this can be spread through the water and feed.
