The 3rd-gen Fire TV Stick and the Fire TV Stick Lite have been rooted. XDA forum user k4y0z has posted instructions for how to root either device, which currently requires the Firestick be on Fire OS version 7.2.7.3 or older.
UNLOCK][ROOT][TWRP][UNBRICK] Fire TV Stick 4K (mantis), Page 111
Rooting an Fire TV Stick 4K » Miguel Mota
Fire TV Stick 3 and Fire TV Stick Lite have been rooted — Custom TWRP Recovery and Custom LineageOS Android 11 ROM available
Rooting an Fire TV Stick 4K » Miguel Mota
How to Flash Custom ROMs on Android With TWRP
Root AFTVnews
כתב אהוב פשוטו כמשמעו twrp android tv box סימפוניה שבועה סחורה שונה
ROM][UNOFFICIAL][11] LineageOS 18.1 for Fire TV Stick Lite/3rd gen (sheldon/p), Page 20
How to Root the Fire TV 2, install TWRP Recovery, and ROM - ALL WITHOUT A COMPUTER
2nd-gen Fire TV Stick hack: Unlock the bootloader, install custom recovery - Liliputing
Root AFTVnews