I'm aware that this used to be possible at one point or another, but I'm trying to find a solution specifically for Big Sur and up. I've recently decided to automatically hide my dock which unfortunately leads to forgetting that I've received text messages until hours later unless I open and respond to them immediately.
Is it possible, either within the system or through third-party software, to add an icon to the menu bar that indicates that there are unread messages in the stock Messages.app?
Messages
Notification style to Alerts. That way the Notfications do not disappear until I dismiss it. (Ofcourse, this is productive only if you don't receive a lot of messages or else you'll spend a lot of time clocking and dismissing each message!)