We are reacting to a tweet that has very little content, notably how high is the bar for psychological harm. It is undoubtedly higher than making someone feel bad, but the question is: is it high enough?
Consider your words, "you are responsible for your own thoughts, feelings, and actions". For the most part, that is true. On the other hand, we also consider some segments of society as vulnerable. In some of those cases they do not have full control of their thoughts, feelings, or actions. Sometimes it is due to a lack of understanding or experience (e.g. children and youth), sometimes it is due to psychological disorders. The question becomes, who is actually responsible in cases where trolls unintentionally or intentionally exploit that disorder?
The other thing must be considered is the magnitude of the infraction. A competent person may be able to handle a random death threat. The person would have to be considerably more resilient to endure an effectively anonymous person stalking them and threatening them. I would question the ability of anyone to handle that sort of behaviour for an extended period of time unless they have the means to prove the threat is not credible or to protect themselves. There is a reason why some people of means go around with bodyguards, and it isn't solely about their physical protection.