I've just figured out how to fix mine. My set up is a MacBook Pro 16 inch (2021), so this might only apply to me.
My understanding was that I could use the Studio Display (nano textured glass version – if at all applicable) to keep my laptop charged while I was using the whole set up. So, I didn't have the MagSafe power plugged in. This morning, I fully unplugged the Studio Display from the wall and MacBook – assuming I'd need to start thinking about returning it. I put my MacBook on charge using MagSafe. Then, I tried the display one last time and it turned on. Assuming it was somehow fixed, I took the MagSafe charger out and put it back in my bag. Right away, the Studio Display turned off. MagSafe back in, the display turns on.
I'm assuming it should charge my MacBook though? So this still feels like a real miss from Apple. At least it's working so I can actually do my work today.
Hope this fixes everyone's issue. Apologies for the novel, I just wanted everyone to potentially recreate the issue if needed.