Re: Duty sex only
She's likely feeling touched out. Young kids are SO exhausting and it's hard for many women to understand the sexual urge their husbands feel when they themselves do not feel it.
I'd highly recommend explaining how you feel to your wife. Start with a very loving "I understand that you're exhausted with the kids and how hard you work with them. Here's what I need as your husband and here's how I think we can make it happen." And present a plan that will help ease the work off her 1-2 nights a week (or even start with 1 night a week if you're ok with that for now, if it's more passionate).
Women with young kids ALWAYS have something to do and someone to take care of, especially if they are co-sleeping.
Plan ahead of time so she knows to reserve her energy. Come up with a way ahead of time to help her with the kids so she can save her energy for you. You take over bathtime/Stories/bedtime routine that night of the week. Let her unwind.
This takes the pressure off since it's only 1 night a week and she has your support, instead of adding another thing to do. I bet she will remember how nice it is to fully relax and engage with you.