The Fire TV 1 running software versions 22.214.171.124, 126.96.36.199, and 188.8.131.52 is rootable, but because those software versions are so old, you have two options.
Option 1: If you allow your Fire TV to start downloading an update, it will very likely begin downloading version 5.0.5 (Build 537174420). That software version is rootable, so if you allow it to update to that version, you can skip several steps in the rooting process. The disadvantage is that you will permanently lock your bootloader by updating to 5.0.5 before you root. This is why option 2 is recommended, but that option involves a very long process. If you don’t care about having an unlocked bootloader, then let your Fire TV update to see what version it decides to download. If it begins downloading version 5.0.5 (Build 537174420), let it finish updating and then continue to this section. If it begins downloading any other version, pull the power before the update finishes downloading and continue to option 2 below.
Option 2: The best option with a Fire TV on such an old software version is to perform a very long process that involves rooting, downgrading, rooting again, unlocking the bootloader, installing ClockworkMod custom recovery, installing pre-rooted 184.108.40.206, installing a boot menu, installing pre-rooted version of 220.127.116.11, installing TWRP custom recovery, and finally installing the latest pre-rooted software version. The end result is a rooted Fire TV 1 with an unlocked bootloader that is running the latest pre-rooted software version. This is the ideal state to be in, but as you can see, it is a long process, which is why some prefer to give up having an unlocked bootloader and going with option 1. To proceed with option 2, follow this guide.