The common-purpose rule has no relevance to flights offered to the public on a compensation basis. These flights are common-carriage operations are subject to 14 CFR 119, making a requirement for an air operator certificate among others, and 14 CFR 135 or 121.
The common-purpose rule is only relevant to operations excluded or exempted by 14 CFR 119 and the pilot acts under a US pilot certificate. A relevant example is a cost-shared flight which is
not advertised to the public. Advertising in online closed groups may be regarded as advertising to the public. See interpretation given 3 Oct 2011 to Haberkorn.
In an interpretation given 21 May 2014 to Andy Dobis:
"It is important to note, however, that the privileges and limitations conferred upon pilots are a separate and distinct issue from whether these flights would be considered a commercial operation for a part 119 air carrier or commercial operator certificate is required."
In the appellate decision made 18 Dec 2015 in
Flytenow Inc. vs. Federal Aviation Administration (
link):
"The common-purpose test has no bearing on whether compensation in the form of passengers' expense sharing, together with holding out to the general public, tends to show that a private pilot is operating as a common carrier."
ChickenHouse, if the common-purpose test is not passed and there is compensation in respect of a non-advertised flight then it is probably a private carriage operation. See 14 CFR 119.23(b).