3 years no sex or touch

maxlonely, welcome to the club nobody wants to be a member of. Keep in mind, you have to live with the outcome of your decisions, even if your choice is to not decide. As such, you need to weigh your options.

From what I have seen, once a marriage has been sexless for two years, the odds of it healing are so close to zero that they might as well be zero.

You are miserable without affection, and you are currently paired with a woman that doesn't want to give you affection. There may be reasons for this, but, my guess is she has not addressed why, so she is comfortable with the lack of sex. If you are at that stalemate, you have five options.

1. Stay, and be miserable.
2. Stay, and push her to do things she doesn't want to do.
3. Stay, and get your needs met outside the relationship with her permission.
4. Stay, but get your needs met outside the relationship without permission.
5. Leave, because the relationship is not meeting your needs and it's the honest thing to do.

Instead of starting with the presence and frequency or quality of sexual activity, consider that to be one of likely several downstream results.
What do each of you think a marriage is? What did you sign up for?
What kinds of things differentiate a married relationship from other close relationships - romantic or otherwise.
Consider: Parent/child, friends, roommates, amicable ex-spouse/co-parent, business partner, boyfriend/girlfriend.
Yes you had a wedding - but what would you both agree differentiates a marriage from those other relationships? Do you have those things?
Would you agree that whatever you have now is not really a marriage, based on what you agree a marriage to be?

If that's the case, where did it go off?
Is it that she knows something about you now or has seen something that she didn't before - something that has made her withdraw to the point that she no longer sees you as a viable sexual partner?
Is it more that she finds herself in a situation (marriage) that she doesn't actually want to be in, so it's the circumstance itself that makes her feel trapped with you as the jailor? Where did things fall off? 3 years ago - did something happen then? Think back to maybe 4-5 years, or even to the beginning of the marriage.

Afraid it's broken record time with nobody's favorite tune: "The ILIASM Two-Step"

1) Obey the Baza.
2) Have "The Talk"

If you divorce, it's her choice.

You will likely always have a connection with her; a co-parent.
She seems to want you to co-parent with her while having sex with someone else.
Is there a reason to wait to bring that about?

She's setting you up for an affair. Perhaps opening teh marriage will force her hand. Much of the punch of an affair is removed when the spouse was aware it was about to happen and did nothing to stop it. It isn't teh story the refuser wants to tell. They want you to have all the blame. If they knew in advance, uncomfortable questions will result.
Yes, you get blame for refusing to be celibate, but people will not see her as blameless. You may be a skunk, but she's "not all there" if she expected non-consensual celibacy to fly. "Ice queen" is not the rep she wants. Refusers want the "martyr" label, and expect the refused to oblige.

If I may, I'd like to tweak ironhamsters chjoices:
"3. Stay, and get your needs met outside the relationship with her permission awareness.

4. Stay, but get your needs met outside the relationship without permission. awareness."

You could throw in option 3a: "3a. Stay, and get your needs met outside the relationship with her awareness and permission/consent/approval.

If you don't get permission, you can do it anyway. 
She can make her own informed decision at that point. Fix things between you, divorce, or let it happen (tacit approval). Ball is in her court. You're fine with any of the three, the first being the Holy Grail.

The problem I have is their mother is not a very competent mother and parent. She’s very loose with discipline and the kids direction. I feel I need to be more present for their sake.

This is why I am holding on.

I have seen people really worse off after a divorce and for all those here and elsewhere that are thrilled with life after that has not been my observation.... 

i do not hold a lot of faith in counseling because finding the right fit or a person that can make both parties see things in different angles is like winning a the lottery can happen but the odds are against it. 
As a person here wrote to me if sex and intimacy is what you expect out of a marriage and that is gone they broke the deal... do not let her make you feel bad for wanting intimacy... that IMO is a way to keep you quiet instead of dealing with the issue.   Mine also wants me to feel bad/ guilty/ selfish for asking for my needs for intimacy to be met. I think it is deflection so they do not have to really think about and deal with the situation...

teens are hard as if they see a difference in parenting and any riff they play it to their advantage .... the more you are focused on your problems the less likely you will see some item they are doing. 

ToughTiger and Maxlonely's observations seem to indicate divorces are pretty unhappy things.

Since we've found 90% of escapees from sexless marriages are happier, it makes me think that couples that have sex find that life outside marriage isn't better than being single. Plausible.

Both could be accurate. Sexless marriages are notably worse than average. Not a tough sell, I'd think.

Maxlonely.
Anything other than the discipline issue?
When it comes to criticizing your spouse's parenting, it's usually the other way around.

Just checking...you don't want to be paternal gatekeeping.
www.drpsychmom.com/2014/07/18/reader-q-a-with-dr-psych-mom-when-will-my-husband-grow-up/

Maybe she, herself, had bad experiences with parental discipline?