I think you’ve received some really unhelpful replies.
It is completely legitimate to want a sexual relationship with your wife, at least after you’ve through with the initial intense period of pregnancy and having babies. Lack of sex will kill most marriages in the long term.
It’s not wrong of your wife not to want sex but in my opinion it is wrong for her not to discuss this with you. I think, if you haven’t already, you should try to get her to have a serious conversation about it - maybe away from home where there’s no possibility of having sex and somewhere private- describe how it’s affecting you, ask for honesty even if she thinks you might be offended, ask open questions and listen. You might do better having the conversation with a counsellor if she’ll agree.
Mumsnet responses tend to focus on whether you are doing enough housework etc. Of course if your wife is exhausted and touched out and resentful she might not feel like sex (and I’ve been there myself with hormonal factors also in play). But we don’t know if it’s the case and it could be any number of other things or nothing she can put her finger on.
For me my ex putting on weight was a significant contributing factor. I never said it to him (perhaps I should have) and knew it was unreasonable on some level but my thought process was that I had worked hard to get my body back after pregnancy etc and he couldn’t be bothered to maintain his, despite having more time to do so.
I personally wouldn’t be bothered by the cycle tracking though wouldn’t want to know you are doing it. I have a far higher sex drive around ovulation and would prefer my partner be aware of that. It doesn’t sound like that applies to your wife at the moment though.
Female desire is mostly responsive. That is not well understood but it means that women rarely (as a generalisation) want sex spontaneously. It’s why some touch throughout the day and gradual build up is important for many women. It also means that if women wait to feel like they’re in the mood sex won’t happen often. Emily Nagoski has written about this in “Come as you are”. If you could get your wife to read that it might be helpful. There are also various podcasts about it.
Sex drive can also be “use it or lose it”. You get out of the habit and feel less need.
I’m not suggesting your wife should force herself if she doesn’t want to. But you could maybe suggest some intimacy without full sex to see how she feels or with a promise you will stop no questions asked and no pressure if she’s not enjoying it.
Also Esther Perell’s “Mating in captivity” is very good on these issues I.e children and domesticity killing desire. It has some stories of couples getting a good sex life back after very long dry spells.
But ultimately there may be nothing either of you can do - this isn’t necessarily anyone’s fault - and you then need to make a decision about whether to continue with a sexless marriage. I think the most people who want sex that isn’t sustainable long-term and even if you think you can make that sacrifice it probably won’t work long-term.