How can I shut off devices without engaging in a tug-of-war?

Yes, there is a global off switch (of sorts). 

• From Your Phone, go to Settings > Screen Time > Select Your Child > Tap DownTime
  
• To shut down your kids’ devices, adjust the Downtime:
  • Set FROM to 1 minute past the present time
  • For example, if the current time is 12:46pm, set Downtime to 12:47pm (it may take a few minutes to kick in)
    

 While there are more elegant interfaces in our opinion, this does work. Just remember to adjust the Downtime when your kids are ready. It’s always much easier to let the technology do the work for you :).