It is not really about that.
It is the anti-money laundering law which they are scared about. The government closing them down when found they are permitting their customers to do transactions that will buy bitcoin as an example.
There is no traces there afterwards.
Once you bought bitcoin they will never know where this money will circulate. It could go anywhere or worse it would be used for illegal things specially creating a terrorist group. 

Try to be positive in every reason and maybe you wont get stress with this kind of thoughts.