iPhone 12 Mini stuck on sleep mode/do not disturb

This has been happening to me too. Once my alarm goes off - the gray "dismiss" has been staying active all day until/unless I restart my phone - but then it begins all over again the next day.

I just looked at the "sleep" icon you posted above. Mine was not in the "on" mode, but when I clicked it on/off the "dismiss" screen has gone away. A little easier than a restart... will see what happens tomorrow. IMO - the changes to the sleep function are problematic. I just want the mellow alarm tones that are avail in sleep to work with a regular alarm. c'mon apple!

Have you tried a hard reset?

https://support.apple.com/en-gb/guide/iphone/iph8903c3ee6/ios

When it is unlocked slide down the control centre and look for the image of a bed, is it turned on? Look at the icons in the image below, it is the one in the lower left. Make sure it is off.

What happens when you use the Dismiss button?