TF2 is a source game which means it cares more about single-core performance than most other games (esp since lol 2007) but having a good GPU does matter. I'm no Setsul but I do not think it will get much better for you unless you opt for using a toaster cfg. I also had stuttering on my gaming laptop because it came with a 5400RPM HDD in it and when I swapped it out for the Samsung 860 EVO SSD that went away.
I would absolutely open it up and replace the thermal paste though before you try anything.