I dont know if my tablet is broken or outdated
OTA system updates won't work if you've modified the system image in any way, including by rooting. To get back to a working state, you need to flash the "stock" ROM, i.e. an unmodified official ROM from Samsung. See Where can I find stock or custom ROMs for my Android device? for details of how to find a stock ROM, and then this existing answer describes in detail how to use Odin to install a ROM on the Galaxy Tab 4.