How can I remount my Android/system as read-write in a bash script using adb?

Probable cause that remount fails is you are not running adb as root.

Shell Script should be as follow.

# Script to mount Android Device as read/write.
# List the Devices.
adb devices;

# Run adb as root (Needs root access).
adb root;

# Since you're running as root su is not required
adb shell mount -o rw,remount /;

If this fails, you could try the below:

# List the Devices.
adb devices;

# Run adb as root
adb root;

adb remount;
adb shell su -c "mount -o rw,remount /";

To find which user you are:

$ adb shell whoami

I could not get the mount command to work without specifying the dev block to mount as /system

#cat /proc/mounts returns ( only the system line here )
/dev/stl12 /system rfs ro,relatime,vfat,log_off,check=no,gid/uid/rwx,iocharset=utf8 0 0

so my working command is
mount -o rw,remount -t rfs /dev/stl12 /system