Can you define what it is you miss about sex?
For many people, that simple three-letter word encompasses a whole range of needs. For others, it is simply a biological release and quite functional.
For people in the first category, it is not just the sex that is missed. It is the feeling of being attractive and desirable to a partner and as a human being. There is also a loss of associated sexual affection, such as passionate kissing or intimate touching and stroking, out of bed. If there is little non-sexual affection too such as hugs and cuddles, there is a sensory loss of touch. There are also psychological needs that are unfulfilled, especially if having sex is associated with being loved and admired. Negotiating sex outside of a relationship that is sought to be maintained can cause difficulties for people in this camp, because there is a great risk that they will fall in love and want to leave the old relationship - and if the new relationship is with someone of the same ilk, that this person will get hurt too.
For people who are easily able to divorce sex from monogamous emotions such as attachment, need for exclusivity and romantic love, sex outside the relationship if openly accessed without deceit, is less problematical, as long as the sexual partner chosen is also of the same ilk and her emotions are similarly containable.
However, bringing a third party into a relationship to prop it up is fraught with human difficulty and is always a risk. Even if people enter into a relationship like this with a promise to contain emotions, they are not always easy to control and that risk is clearly mathematically doubled because it requires two people to keep their emotions in check.
I'm hoping you are more ethical than to consider paying for sex, so hopefully you don't consider that a realistic option.
If you still love your wife and desire her - and you are clearly good co-parents and companions - I think your best bet is to have a different conversation. Not in a threatening punitive 'I'll leave you if this doesn't change' way, but in a manner that communicates your unhappiness and which points out that if she loves you as much as you love her, she might want to help you address your sadness and how this is making you feel as a person.
Do some research on counselling (including pyscho-sexual) and if the conversation runs aground as before, ask her to go with you because it sounds as though you need help communicating, apart from anything else.
I understand why your wife might baulk at the 'what's wrong with you?' approach because her defences will go up and one of those is clearly to believe that the situation is 'normal' for couples at your lifestage. A third party who can dismantle this with skill, without making your wife feel abnormal, might be very productive.
Some of your wife's possible other issues are more easily remedied. If you aren't pulling your weight with the children or housework and your wife is expected to remember everything about the way the household runs and important family dates, this can often lead to a mother-child dynamic, which is the enemy of sexual attraction. If you have ever been guilty of effecting an air of learned helplessness about any of these tasks, make some changes and introduce more equality into your relationship. Listen more than talk. Try to find out your wife's hopes, dreams and aspirations and encourage her to think of herself as an individual and not as a mother and wife. Just those simple but basic changes can promote more individuality in your wife and encourage her to think of herself as a sexual woman, just as she might have done before settling down.