I don't have a usable personal opinion or a prediction about the case.
But from a historical perspective, I think the "holding out" question is more about public perception than profit motive. The original common law concept - going at least back to the stagecoach - was the way one differentiated a common carrier from a private one. One of the early differences between them was liability. A common carrier's liability was the earliest form of strict liability. There are cases on it at least as far back as 18th century England. The strict liability was was based on a legal expectation that, once entrusted, the common carrier had to meet the obligation to carry the person or the goods and get them to the destination safely. "Holding out" the availability of services to the public was just one of the factors to show whether or not the transporter was a common carrier or not.
The guy who is shipping a package or traveling probably doesn't know or give a crap whether the transporter is making a profit, wants to make a profit, or simply defraying some costs for something he likes to do. Maybe that 1800s stagecoach operator likes to travel around the country and figires, "if I take a passenger and some goods for sale, it will cost me less." The customer just wants to know that he's got safe, reliable transport.
The question is ultimately whether he should. What duties, responsibilities, and regulation should flow from the relationship? Those are (and always have been) questions about social objectives. Profit motive is typically a bad guideline for that, although it definitely plays a part - the modern limitations of common law carrier liability are mostly regulatory and, I'll take a wild guess, were primarily supported by the carriers.