OP here. I felt like he started it and he felt like I started it. I sort of wish we had it on video so we could review it and see how it actually developed--it's very hard to reconstruct objectively after the fact when both people are feeling wronged.
OP, this is telling (and I've been exactly where you are, with the above thought). I think you guys are stuck in a bad pattern about who is right, whose needs come first, who is being heard/not heard, etc. I suspect you're somewhat controlling, although you might go to great lengths to deny that. Think your DH is probably very sensitive to perceived criticism, too but I have less to go on there.
anyway: this is the thing. You're both stuck--stuck in this stupid fight, stuck in the unresolved nature of it, stuck in how you unproductively deal with it. Its a pattern in which neither of you feel heard or valued. You are resentful, and make what your DH thinks are angry accusations--he shuts down at your anger and criticism, you feel unheard and continue to carp. You both feel wronged in this relationship, and there is truly no point in trying to figure out "who was right" in this situation. This is not black and white, it is about how each of you feels unvalued, unheard, unappreciated, etc.
I will wager this: you feel unheard when you express that you feel you shoulder too much of the burden--in fact, by focusing on your anger, your DH avoids that responsibility. But your DH feels unheard too--he wants to be able to engage without constant anger and criticism. The fact that you say that there should be room for both of you to have your feelings, but then you interpret his walking away calmly from the argument, with no ill effects on you or your kids, as invalid belies that. The fact that you equated his walking away from an argument calmly as a sign that he is going to abandon your family also says that you believe, deep down, that there's only one way to have and handle emotions, and that is your way. To me, the fact that he walked away would be an indication that he really could not tolerate the way that the two of you fight this recurrent fight, and he wanted to stop it and try something else. this is a sign that he, at least, sees how unproductive the fighting is. You may take it as abandonment or in the very least avoidance of the issue. Neither of you is right, or wrong, in any black and white way, but the focus on who is right or wrong here is a sign of bigger problems. For example: your defense of fighting in front the kids is a defense that you think is valid (fighting is a good thing if we can resolve it peacefully) but his walking away to take a break is a defense (fighting a stupid fight in front of kids is not good, better to engage later) is invalid for you.
Counseling, the right counseling, can do wonders, but only if you can drop your defensive mechanisms and truly listen to what your husband is trying to tel you about the way you talk to him and treat him. Listening to him and accepting that does not invalidate your own feelings at all--but I think you're operating under the assumption that if he's right then you're wrong, and vice versa. I think you're way too hung up on this right/wrong dynamic and it is definitely not good for the health of your marriage. The fact that you wish you had a video to prove who 'started it' is in itself kind of pointless. You both engaged, until he walked away.
Resentment over household chores with two young kids can be very, very draining and stressful. DH and I went through a very similar dynamic. But when we went to counseling, we learned not how to divvy up chores, etc, we learned to communicate and to get out of these patterns, I don't think either one of us realized how much resentment and avoidance we brought into our daily interactions or how much general tension there was, even if we didn't fight. Counseling for us required us to really examine our own acts and speech and not be reactive so much as empathetic. For me, learning that if I validated Dh's feelings, it didn't invalidate my own conflicting ones, was huge. It allowed me to see that I could be empathetic,andnot critical, without 'giving in'--that there was room for both of us to be right in how we felt, which allowed us to work more productively as a team to solve our differences.
So I suggest that you put aside the specifics of who is right/wrong, who is doing more, etc. Not that this isn't relevant, but until you actually learn to hear and validate each other's feelings then you can't solve the logistical issues. And once there is trust and acceptance established, then its more likely they will be resolved without conflict.