It's important to understand that the Soviet intelligence apparatus and leadership didn't necessarily understand on a fundamental level how power transition worked in the United States. In the USSR, elections were held for show, and actual leadership was determined by behind-the-scenes power struggles. So it was difficult for them to grasp on an intuitive level (even if they knew it intellectually) that the constitution and elections actually matter in the United States.

So from their point of view, when President Reagan was shot, it was only natural for there to be a power struggle and when Haig went on TV and declared "I'm in control here," it would seem to confirm their suspicions.