Yes, yes and...yes to your questions. Did you try replacing the fob battery? You ask a question, and then question the answer. It will cost you a $2 battery to try it. The fob transmits continuously so the vehicle knows it is OK to unlock the doors for you. The doors will not unlock if they do not sense the programmed matched fob. There are multiple short-range antennas in your GC, including one around the perimeter of the spare tire well (for tailgate operation). The fob will start operating randomly as the battery gets weak and then eventually not at all, requiring you to use the fob manual key to unlock the door. And possibly having to remove the dash pushbutton and use the end of the fob to start. There are many discussions of this on the forum. Do a search if you do not like my answers.
You ever notice how your TV remote control buttons start to require multiple pushes, or operate poorly when the batteries get low? The difference is your TV remote (and old style passive automotive remote controls) only sends a signal when you push a button. The GC Enter-n-Go system fob is always sending a signal.
How old is the OEM battery in your fob? How long was it in the parts bin at the factory? How long from your GC manufacture date to today? The OEM battery started using and losing power the instant it was installed in the fob at the factory.