Hi, OP. Rose asked me to take a look at this thread, as my husband has travelled for work/lived away most of the last several years.
I understand where the folks who are telling you you need to be home are coming from. But my experience is that if your head is not in the game, and into meeting her needs while at home, it really does not matter much if you are home or not. Iow, if you are a 4th kid, keep the job you have, as it will be less work for her. Sorry if that sounds harsh.
A big difference between your wife and me is that I have always wanted physical and emotional intimacy when my husband has been home. I don't really understand this once or twice a month sex idea. I have heard that some women get really into their kids and lose interest in their husbands. Could be the case with your wife. She does seem super involved with the kids.
My husband is not one to be resentful if for some reason we do not have it every day he is home. He initiates, and I usually respond. But if I don't, it is no big deal. He just kisses my forehead and we both go to sleep. No resentment.
If you are needy or whiny or pouting or anything like that, stop. That is surely unattractive. Just try to inspire her interest, if possible. No guilting allowed.
Just thinking about this more . . . my husband can be pretty nurturing. And even when he is not nurturing, he is not demanding or needy. So between giving to me, and not taking on a personal level from me, it allows desire to build on my end. The last thing a mom of little ones needs is a husband taking from her.
Do you really like your job? I would hesitate to quit if you really like it, especially if you are unlikely to make as much money at a local job, and if your wife is not in agreement with the change.
Good luck, OP. I hope something in here helps. And remember: kids are not little forever. This too shall pass.