I wonder if using the “assign duplicate tasks” might work for you - you can put the prompt together and then assign a duplicate task to everyone on the team member at once. You could also add the recurrence if you want them to answer this question daily or weekly. If you want the answers to be “visible” to everyone- make everyone a collaborator on the first task before you assign it to everyone.
Hmmm… not certain. I think if the task was in a project that everyone was a member of - they might also have the visibility. You could do a test task to see (add just one or two people and see if the third person can still view the responses).
Edit to add: If you do have to add everyone, the nice thing is you can just do that 1 time and make it a task template!