I agree with you about the telling people, & I think most people will have more respect for someone who does. I think there is still a bit of a stigma in some people's minds though - we know strangles is simply a bacterial infection but there's probably a few people around who see an infected yard & think it's badly managed or dirty - and so people are reluctant to disclose it.
Remember the horses in the field have tested clear, so even if a bit of hay they've pulled off & dropped back into the truck should blow into your field & then yours should chance to eat it there's no problem because there's no infection to transfer. And are your horses really going to be falling on a bit of loose, blowing around hay when they have enough to eat & the grass is starting to grow?