I mean, the Thalmor will certainly come after you when youre just some rando too. I'm not 100% certain but I believe the trigger to start getting the assassins sent after you is just to kill a Thalmor Justiciar or other Thalmor character. Presumably all Jarls look askance at random murder (that they didnt sign off on, looking at you Siddgeir) due to it being, you know, murder.
Which is the point. The Thalmor aren't judge, jury or executioner. Theyre either advisors or murderers.