I don't think it's the ad, per say, but the agreement for delivery. It's be like you sending housing ads and asking USPS to only deliver them to men. USPS would probably be on the hook then as well.
Back to your point though, it is a new domain because with your paper examples there is no one who can "undo" the illegal ad. You copy it at Kinko's and it's not like Kinko's is out there delivering it with you. With online platforms, they both Target and deliver these ads and are responsible for continually delivering them.
Example: I worked at a pharmacy with the self-serve digital photo services. A customer brought in a CD and made christmas cards from a family picture.
It turns out the picture was taken by a local photographer the customer had hired. The pharmacy I worked at got the legal notice. Supposedly, we were partly to blame since we profited off of the cards the customer printed. Nevermind the fact that we didn't generally check the customer's pictures, nor could we actually have guessed these were professional vs a good picture.
The customer cropped out the identifying marks. We had all seen pictures that looked professional, both in digital and film form. It seems none of that mattered. It wouldn't be Kinko's fault that you photocopy an illegal ad, but they can be party to the fault by allowing it to be sold.
Now, the USPS probably has completely different laws protecting it. I'm going to guess it is more likely that they try to get you for mail fraud rather than actually hold the USPS responsible.
If you get photos printed at Walmart (or at least when I tried a few years ago) they make you either sign an indemnification agreement or refuse to print the photos. I remember one issue in particular - my wife had purchased some logos/prints online from someone on Etsy for a friend's baby shower. They still had the watermark on them and Walmart refused to print them without a signed release from the original artist.
That's common on the software from the companies producing the machines (we had it too), and isn't quite enough.
It is pretty ridiculous in these cases, honestly. This particular picture didn't have a "photographer's studio" setup - it was an outdoor picture with the family around a long, metal farm gate and the photographer sold the customer the CD. There wasn't much the store could do to prevent this, and we had to start physically checking a lot of pictures at that store.
In the US and many other countries (I can confirm this is true for Japan), pharmacies are also convenience stores, and sell many other things besides drugs.
I did once hear that part of it is historical. That in some places, many years ago, because photo developing was a much more manual process using more primitive means, and given that pharmacists of the time were used to handling and storing chemicals, and carefully following chemical processes and so on, that there was a natural link between the two. This would be back before the time of posting your film to Snappy Snaps, or there being a centralised photo-developing laboratory with an associated chain of collection huts that simply acted as the interface to the public. Once an association is created - exposed film goes to the pharmacy to be developed - it tends to linger.
However, I've not been able to find any evidence of this. It could just be a just-so story.
Back to your point though, it is a new domain because with your paper examples there is no one who can "undo" the illegal ad. You copy it at Kinko's and it's not like Kinko's is out there delivering it with you. With online platforms, they both Target and deliver these ads and are responsible for continually delivering them.