Well,there's a difference between Thalmors and Altmers.The Thalmor's more or like the Nazi government of Germany and all their cronies while the Altmer's more like the German race.Just as there are a lot of German Americans,there's also a lot of Altmers who are imperial citizens living in the mainland of Tamriel.The Dovahkiin,even when he/she's an Altmer,is described to be an imperial citizen living on the mainland.You don't get the option to declare that you are actually from the Summerset Isles.
Altmer being catched while illegally crossing border. Conveniently meets Ulfric Stormcloak. Totally not suspicious. No wonder imperials just decided to execute everyone. After that he runs from Helgen with Ralof, known Stormcloak. It don't even require much paranoia to connect the dots. Just that Tullius apparently didn't read the wiki and hence didn't know this was typical Elder Scrolls protagonist coincidence and the Dovahkiin is totally innocent. What he do know is that the Thalmor are masters of deceit and he is convinced that their undercover work in Skyrim is extensive and they are supporting Stormcloaks, however indirect it may be. Furthermore, he is probably right.
And about Thalmor and Altmer. Not every Altmer is Thalmor, but at least in Skyrim every Thalmor is an Altmer. At least official Thalmor, i somehow don't believe they haven't got agents among all other races.