Since you only have 1 remote...before tearing into the car's side of things...stick with the simple thing first...and that's the key.
You've already discovered the battery is ok...now (as previously suggested) make sure the 315 MHZ RF signal is being transmitted from the key before tearing into the car's FZV (central locking system) & DWA (anti theft alarm siren system). Find a local place (usually an auto parts store) that has a remote key frequency tester
BTW, the components for the FZV/DWA systems are similar but in different locations depending if you have a sedan, Touring, or SAV...so don't assume when reading info from a "sedan" forum that it applies to the X5. For example, the receiver in the E53 X5 is located above the tailgate' window...and the antenna is in the left rear cargo window (same antenna location as the Tourings). And though FZV & DWA are controlled by the GM and the signal from the remote key travels through the same components...when it reaches the GM...commands are the split & sent to central locking components to lock/unlock the vehicle & DWA components to arm/disarm the alarm siren system: