The CCL is the default and by far largest NFP group in the US. They run the majority of diocesan NFP programs. Given that they are the biggest face of NFP in the US, I hardly think that it’s reasonable to call them extremist: they
are mainstream NFP for Americans. They’re welcome to their points of view, certainly, but to teach that only stay at home mothers are fulfilling God’s will, that every woman’s body functions identically and perfectly, and that if a mother can’t fulfill an unreasonable and unachievable set of standards (see: napping with baby while a toddler runs loose) she’s just not holy enough is wrong, and spiritually abusive.
Anecdotally, the Creighton/Napro groups (second largest group in the US) are just about as bad. The only official way to get new charts is to pay an instructor for them, and the only way to do that is to pay $50+/hour for a consultation on top of the chart prices, which seem to be set by the instructors. Bear in mind that the charts themselves are simply photocopied black-and-white charts, but instructors will often charge $20 for a single piece of photocopied paper. If you think that’s silly and simply copy one of your own before using it, you’re told that that’s “achieving behavior.” Apparently, using your personal copy machine rather than an instructor’s makes your chart far more likely to result in pregnancy.

That’s leaving aside the fact that their default answer for problems in charting is “see one of our doctors”–who only accept one or two insurance plans, are sometimes not even covered by those, and frequently require many, many visits at many hundreds of dollars apiece.
I know much less about the Billings method and don’t know anyone in real life who can comment on it, so I’ll not comment on it. I personally have used STM (the method promoted by CCL) and now use Marquette (a combination of hormone testing and mucus observation), and while I’m also happy with Marquette because it fits my body and personality best, I like the blunt honesty I’ve gotten when talking to the method’s developers. I like not being lied to. I like having a medical provider say “yes, abstinence is hard, you can anticipate it lasting approximately this long, stay strong, you can do it because it’s worth it” rather than gaslighting me.
One last consideration: the orders of religious that are growing rapidly now are ones which have a harder standard of life than others. Much like the Marines, they’re honest with their “recruits,” and much like the Marines, they end up forming strong (spiritual) warriors because they’re tough but honest. Their postulants are told up front that their lifestyles will involve fasting, abstinence, long hours of prayer, interrupted sleep, and in the case of the active orders, service to others. They
aren’t told “no one here likes meat anyway,” “we all enjoy only sleeping four hours at a time, if you don’t, it means you’re a bad person,” or “there’s nothing quite so fun as changing dressings on an end-stage cancer patient’s sores.” They are told “it’s hard, but it’s totally worth it in the end.” And they’re right.
I think we’d all be well-served if NFP instructors and marketing, and most especially priests and deacons in their preaching, took that approach to the vocation of marriage, too–that it’s often hard, but it’s also worth it, and that anything worth doing is going to be hard sometimes, whether that’s periodic, extensive, or total abstinence, the inevitable marital disagreements that come up, parenting lots of small kids or teenagers, and so on.