This is contestable. I wrote a longer reply above but where I would say this is contestable is in question of what the state's interests actually are. I think you're operating on the assumption that the state's interests are that of efficiently allocating public resources and maximizing the safety and wellness of its citizens. Something most people probably want from their governments. But this is not necessarily the state's own interests. I don't think that many would argue that many members of the state actively encourage inefficiency, especially when they can personally benefit from such inefficiencies, even if at the expense of the citizens they represent. You know, treason. "Just a little treason"
Fwiw, I don't think this distinction takes away from your point, which I agree with. It just adds to potential complexity needed to possibly find a solution.
But that fails to the original point -- crossing the border with a physical medium in your possession isn't necessary to export the information out of the country, so you cannot prevent it by conducting searches there.
The state has no legitimate interest in measures that are ineffective.