They're not arguing that they have the right to publish ads for illegal services. They're saying they're not liable for what 3rd parties post on their site.
They're saying they're more like journalists, simply reporting what other people are saying (which happens to be promoting illegal services). While it sounds like they are pro-decriminalizing sex work, no one is arguing that it's legal to advertise criminal behavior (selling drugs, weapons, sex, etc.). They're just telling the courts to go after those advertisers individually rather than Backpage.
They clearly wanted to profit off of the sex industry in places where it was still illegal to do so.
The courts and the congress have a very strange interpretation of this, for example in: https://en.wikipedia.org/wiki/Gonzales_v._Raich
(note: I am not saying I disagree with a decision to outlaw discrimination, just pointing out the absurdity that the Commerce Clause was 'interpreted' as being the only way to do this [which ultimately resulted in government limiting free speech])
https://en.wikipedia.org/wiki/Wickard_v._Filburn
In hand-wavy terms... in this case the commerce clause was used to prevent a farmer from growing wheat on his own land for use in feeding his own livestock. The federal government had established limits on wheat production at the time as part of the "New Deal".
The "time of war" argument is completely bogus. The word "war" doesn't appear anywhere in the decision itself nor are its consequences limited to wartime in any way. The law in question was a 1938 law that was trying to increase government management of agriculture economic activity. While progressive are happy to use any crisis that comes along to further this sort of overreach into economic activity, including war, one could equally say that the courts decision was probably at least as influenced by the fact that Roosevelt had appointed 8 of the Justices that heard the case and the court was ideologically aligned with him... since the 1938 law was part of his disastrous economic program (http://newsroom.ucla.edu/releases/FDR-s-Policies-Prolonged-D...), it's little wonder they sided with the government.
And wheat absolutely was a national, not a local, product in 1930, which made it well within the textual scope of the commerce clause. Whether or not they should have implemented wheat controls is separate from the fact that they had the power to do so.
The whole reason this power was given to Congress was so that no one state could unfairly regulate economic activity in another state. The idea being that it was better to have one set of relatively neutral rules than dozens.
As for your study: the two authors of the study are noted neoconservatives who claim that the economy only recovered when unions lost their power, completely at odds with actual history, in which the height of the American economy coincided with the rise of union power.