To unmute regions, use the Mute tool (from the Toolbox) —not the Mute button on the track. You can also select a Region and press M on the keyboard to mute or unmute it.
The benefit of this option is that it allows you to comfortably record take after take without starting and stopping Logic. After the take, you may mute and unmute the Regions individually to compare and composite the different takes.
If you don't want Logic to automatically mute Regions in cycle record, uncheck The Auto Mute in Cycle Record option in File > Song Settings > Recording.