> My understanding is they're arguing that the FCC didn't have the authority to regulate net neutrality,
The fight to preserve net neutrality rules was actually (and continues to be) a fight to preserve most of the FCC's ability to regulate ISPs at all [1]:
> On October 19, 2023, the Federal Communications Commission (FCC) adopted a Notice of Proposed Rulemaking (NPRM) that proposes to reclassify broadband internet access service (BIAS) as a Title II common carrier service and reinstate net neutrality rules.
...
> The FCC’s ability to adopt net neutrality rules depends on the legal classification of BIAS under the Communications Act of 1934 (“the Act”). As amended, the act defines two mutually exclusive categories of services: telecommunications services and information services. While telecommunications service providers are treated as highly regulated common carriers under Title II of the act, the FCC has much more limited regulatory authority over information service providers.
The FCC under Ajit Pai changed the classification of broadband providers from Title II ("information services") to Title I ("telecommunications services"). The FCC under Jessica Rosenworcel is trying to change broadband providers back to Title II, which (among other things, I assume) would allow the FCC to treat broadband providers as common carriers. The FCC might be able to regulate rates (meaning prices, not bandwidth) for internet service if ISPs are common carriers, but I'm not completely certain.
The 6th Circuit court case mentioned in the last section of the article is large ISPs' attempt to argue that the FCC has no authority to treat broadband providers as Title II services. Hence, the last section of the article is called
> FCC must show that broadband is telecommunications
The main court case of the article is about a New York state law which would regulate rates for internet service for low-income households. The FCC can't regulate rates without Title II authority. (Relying on the 10th Amendment, I assume) New York believes that the states have the authority to regulate ISPs in such a way (barring constitutional restrictions such as the Dormant Commerce Clause or the First Amendment). The large broadband providers are trying to petition the Supreme Court and argue that the FCC's Title I authority over broadband providers preempts states' attempts to regulate broadband providers in ways that the FCC would only be able to do with Title II authority. If the ISPs' argument prevails, the outcome would be that both the FCC and the states would lack authority to regulate rates on internet service, treat ISPs as common carriers, or impose net neutrality rules.
[1][PDF] https://crsreports.congress.gov/product/pdf/IF/IF12513