I think the "incomplete" or perhaps the more accurate "false" story is that the FCC chairman threatened to pull licenses of ABC affiliates who continue to broadcast Jimmy Kimmel Live.
I saw the chairman's entire statement which was a response to an apparent interview question which I unfortunately haven't seen or heard. His response was that if ABC/Disney's affiliate customers objected to the content of ABC's product, they should let ABC know and perhaps consider not carrying whichever content they found objectionable. He then went on to muddy the waters somewhat by expounding on the FCC's responsibility to ensure networks and affiliates were adhering to their licensing agreements which include the responsibility to operate within the "public interest, convenience and necessity". I think he then went on to suggest that purposely providing inaccurate information such as Kimmel's allegation that Kirk's shooter was MAGA was a direct contradiction to the public interest, convenience and necessity rule. He did not threaten to pull the license of any affiliates broadcasting specific programming.
Now, Sinclair and Nexstar were apparently listening and decided to do exactly what his hypothetical answer suggested, they decided to stop broadcasting one of ABC's products until such time as they deemed that product reflected the values their customers expected and their advertisers were willing to support. While part of that decision may have been an effort to curry favor within the FCC, that was obviously a business decision, not a political one.
That's my take anyway.