I don't really know what to add here, but I backed up Team Fortress 2 when I was still using Manjaro. I moved to Ubuntu and I restored it. Now, it won't open. Here are the logs: https://pastebin.com/aRFCZZQ4.
There are two lines that are quite interesting inside your logfile
ERROR: ld.so: object '/home/icebunny08/.steam/ubuntu12_32/gameoverlayrenderer.so' from LD_PRELOAD cannot be preloaded (wrong ELF class: ELFCLASS32): ignored.
/bin/sh: 1: /home/icebunny08/.steam/steam/steamapps/common/Team Fortress 2/hl2.sh: Permission denied
First one seems like it tries to load a 32 bit libary into a 64 bit process.
The second one: you don't have the permission to the file located at
/home/icebunny08/.steam/steam/steamapps/common/Team Fortress 2/hl2.sh. Either it don't exists, or it can't be executed, or you just don't have the permission.