I was the one that migrated the question. In retrospect, I probably should have left a comment as to why I made the decision, so for that, I apologize.
You asked two sub-questions, and I will address them separately.
Why was Wunderlist designed to not work in private browsing mode in Firefox?
This first question, asking about the company's reasoning/design decisions for explicitly not allowing private browsing in Firefox is, to me, something that only the company can answer. From that perspective, it would be off-topic since those answering could only give opinions as to the developers' intentions. If we instead consider the question from the perspective of potentially being a technical limitation of Firefox's private browsing mode, it is my opinion that the experts on Super User would be best equipped to address that.
Is there a known workaround for this, by tweaking the setting for instance?
The second question asked seemed to have less to do with the use of the app itself, and more to do with why it wasn't working with particular browser settings, which is something that I would say is an issue for Firefox troubleshooters on SU, more than a usage issue specific to Wunderlist. If you meant how to change the Wunderlist settings on their site to accommodate this, that question would certainly be on-topic here.
On Web Apps, we tend to be more about end-user questions on the web applications themselves, and less about the infrastructure and technology that they run on. While I think your question teeter-totters on the edge of our scope, I had to decide where it would likely get the best answer.
If, after considering my opinions above, you believe that Web Apps is the best location for the question, I support SU rejecting the migration, or, probably more practically, simply deleting it there and re-asking it here.