Unable to adb push on Samsung S4
If the ADB daemon is not running in root mode, you cannot push to /system/app even with /system mounted read-write, due to not be given the permission. As you correctly pointed out, adb root is no option on non-development-devices ("adbd cannot run as root in production builds").

To fix this up, take a look at chainfire's adb Insecure. This app requires root (a condition your device obviously meets), and (temporarily) patches the ADB daemon to run in root mode. So with "insecure mode" enabled, you should be able to push to /system/app.

Alternatively, push to e.g. /data/local, then login to the device with adb shell, execute su to switch to root-mode, and move the files (mv /data/local/whatever.apk /system/app) while /system is mounted read-write.