10 days per cycle is pretty good.
Bear in mind that an average of twice a week (which is what 10 times per cycle probably averages to) is pretty good, NFP aside.
This article says that married couples age 18-29 have sex 109 times a year (i.e. 2X a week) while 60 somethings have sex 32 times a year (i.e. roughly every other week).
articles.chicagotribune.com/2010-12-15/health/sc-health-1215-how-often-sex-20101215_1_unmarried-couples-relationship-sexual-problems
uhhhh no. It is a common “sell” used by pro NFPers that 10 days per cycle means 10 days for sex. Nope. It could very well mean no days of sex at all in the entire cycle. Especially if the woman is not feeling well, not in the mood, busy or even if the kids are sick. A ten day window once a month is not a great window. Especially if 5 of those ten days are menstruation days and the couple is not super into that.
When we used NFP there were some months it was once, maybe.
I can speak directly to the OPs issue. My wife was not open to life as much as I was. I begged her to get off ABC and use NFP. I begged her to be open to life, to want a large family. As her faith grew so did her understanding and heart. I would make deals with her, ok we can avoid for now but we need to revisit this after (fill in the blank) My wife would try to control the exact right time to conceive another child and the exact right number to have. There was no serious reason to avoid and we found ourselves sating things like “well, after Christmas.” Well, after we go to Italy for a vacation, we don’t want to be pregnant in Italy!" See, the reasons were not “just” for both of us. It was our 4th child when we actually made a decision to “try to conceive” It took 3 months. Since we abandoned NFP our lives have been so much better. No charting, (though you never really abandon it after you learn it.) My wife is well aware of her fertility and so am I. But it is not something we argue about anymore. We have decided to live as a family, as husband and wife. And should no kids be in our bed for a night we are free to either watch a show and fall asleep, or attend to one another in the marital embrace. No, computer, no charts, no signs, no rules and exceptions, no calls to the C2CL to figure out our charts as foreplay. It is nice. The OP’s wife is at odds with the OP and it is affecting the marriage. She is either doing a poor job of communicating her faith and troubles with more children, or she is just not there yet in terms of her marriage or her faith.
Either way the answer is not just “well men want more sex so he must be doing something wrong or needs to change his view” The answer involves both of them, communication and God. From the posters words and from my own experience it is not just that he wants to have sex more, but rather he needs that form of manifested love inside a marriage. That love is not artificially satisfied by handholding in the park or breakfast in bed. It is satisfied the way God intends, through unitive and procreative action. He is starving, and she is on a diet.

A situation at odds like that will not continue to be healthy.