I haven’t noticed anyone saying there is no cost. What is being said is that the benefit outweighs the cost. Without a tax increase, the money for extended benefits increases the deficit, which is costly - but worth it in this case.
They are all forms of stimulus. Hell, tax cuts for the rich are a form of stimulus also - just not a very good one. Some try to create jobs, some try to preserve existing jobs, like payments to the states to keep cops and teachers from being laid off, and some try to maintain consumption, understanding that the government is not going to fix the job market through any kind of stimulus package. It sounds like extending unemployment benefits is a fairly effective form of stimulus. Beats breaking windows.
Good - and the study cited is a way of measuring this.
Notice we don’t even have to get into the human costs of long-term unemployment to see that it is a good idea.