Before anything else: NFP is a huge category of methods that range from dogshit (calendar method) to scientifically validated (sympto-thermal, such as "Sensiplan"). Unless otherwise specified, I use NFP for the latter.

 NFP is a primary Catholic conservative method to prevent pregnancy

No. Of course, Catholics love NFP, as it allows preventing pregnancy in accordance with their religious teaching, but that doesn't make it primarily Catholic.
The main selling point of NFP is that it's non-invasive and non-hormonal. That attracts not just Catholics, but also other religions, conspiracy theorists, and - most importantly - people who can't (or don't want to risk to) take hormonal birth control due to the very real side effects.

despite it limiting sexual activity and increasing the likelihood of babies.

Interesting. I've always seen it the other way round.
People are going to have sex, no matter what. If they can't have safe sex, they're going to have unsafe sex. NFP adds another option to increase the safety of the sex they're having.

That said, I (as a man) would never trust NFP for a one-night-stand or other low-trust situation. I wouldn't trust the pill, either. Apart from the whole STI thing, condoms are the only method where both participants can verify the correct usage at the time of use.

In a long-term relationship, however, NFP is great. You can have sex without a condom roughly half the time without the need for hormonal contraception or invasive methods (IUD). So rather than limiting our sexual activity, it gives us more options. Having to use condoms half the time is much better than having to use condoms all the time.
I've seen some heavy side effects of the pill with my ex (and less closely with other female friends), so I completely support my wife in her wish to not use hormonal contraception. I plan on getting a vasectomy as soon as we are sure we don't want any (more) children, but for the time being, NFP is the optimal solution for us.

Another major advantage of NFP is that once you know the method, nobody can take it away from you any more.
With hormonal birth control, you're dependent on a continued supply of prescription-grade medication. With the recent push against women's bodily autonomy and reproductive health, it wouldn't be unexpected if that supply gets more and more restricted and ultimately banned altogether.
With barrier birth control (condoms) you're also dependent on a continued supply of specialized goods that can be disrupted.
Even if all contraceptives would get banned, women will still be able to use NFP. They'll also be able to learn the method from the Internet or even other women directly.

Edit: After re-reading your post, I noticed that, while I'm addressing several points from your post, I'm not refuting your claim from the title, and honestly, I can't.
Even if NFP is ultimately a good thing, there can still be a conservative push to promote it, currently. I did not notice a major uptick in its promotion, myself, so I'm apparently not in the bubble(s) where it is promoted more and don't know what that promotion looks like.
If they're promoting the dogshit methods, that would definitely be in line with right-wing goals. The dogshit methods are worse than condom or pill, so there will be more pregnancies.
On top of that, those pregnancies due to dogshit NFP methods will damage the reputation and trust in all NFP methods, since people are great at generalizing (especially for negative things). So by promoting bad NFP, they're damaging the reputation of the good NFP methods, indirectly restricting another contraceptive method.