So far this sounds accurate.
Wait a second here. The Null hypothesis only establishes the burden of proof in a limited formal sense. There is also the issue of precaution. If someone claims "if you eat that, you'll die," that person will not have to provide much evidence if any before a reasonable person will hesitate to take a bite.
And then, the article becomes sort of 'alarmist' when it comes to a cooling climate. So, that whole discussion of what is or isn't the null hypothesis becomes irrelevant if you are proposing two different potential relationships, rather than the presence or absence of a single proposed relationship.
There are rational reasons to be concerned about a warming climate. There are rational reasons to be concerned about a cooling climate. The difference I would guess is that we have already seen climate warming going on already, and its related problems, and have good reasons to believe that human activity is a large contributing factor (though human activity in the pre-industrial age leading to warming and cooling periods can't be ruled out either). Whether we need to be concerned about cooling in the medium term is much less clear, even if there is some suggestion for it based on past long-term trends.