Well, I'm going to come at this from a different direction - and I'm probably not going to be very popular for my opinion.
Your wife is a stay at home mom. It's her job to take care of the house and the child. If she wants, she can find a job and you can become a stay at home dad. Or you can both work and split household and child care evenly - in the last two situations she has the right to complain. In the first, none. I'm not saying there isn't value in a stay at home Mom, and I'm not saying it isn't hard work - but that was her choice.
Right now it's her job to take care of the children. You help her out? Great, you're a great father and husband. But to endanger your job by taking your phone because she feels resentful, or that you're not helping enough? It's her job. And by taking your phone, she's threatening your ability to allow her to continue to stay at home and provide superior care for your children. It was stupid, childish, immature, and frankly, I'd have been very angry if it was done to me.
Your wife needs to back down and re-evaluate what you provide for her, and she needs to re-evaluate how she values the time she's given to spend with her children and bond with them, rather than dropping them off at a daycare on a daily basis while she goes off to a cubicle to deal with even bigger morons.
And having young children is nothing. Everyone has children. Everyone has family. Expecting your job to treat you differently because you have young children is insane, and only leads to bias against other parents.
My wife was a stay at home mom for thirteen years. We never had a conflict like this. She breastfed both of my children - one until 19 months. This was a burden for her, but we made the choice together to do so. Anytime I was needed for work she let me go. She slept often on the couch or futon when the kids weren't feeling well to avoid disturbing my sleep. I'm in IT as well, and I'm also on call 1 week a phone (and backup on call a 2nd week of the month). And I have to do disaster recovery exercises once every two months. And I'm salary and often placed on projects with time crunches, leading me to work long hours. How much time do you think I got to help with chores or kids?
One thing that may factor in is that many people feel that IT jobs aren't "real work." Perhaps your wife feels this way. That they don't mentally tax you or exhaust you like a "normal" job. This perception needs to be taken out and shot. When I have a system down that generates millions of dollars in revenue an hour, I'm pretty stressed. When that situation ends, I'm pretty tired. If I have a deadline I'm not going to meet, I feel the same pressures and stresses as a construction worker. Is my job physically demanding? No. But it is extremely mentally demanding.
TL;DR: Your wife needs to shut up, suck it up and accept what her job is, or find a job outside the home and you need to start splitting all chores evenly. (Income differential does not allow you to take a smaller percentage of chores. Work is work.)
If she ever interferes with your work phone again, well... that'd be a fight in my home. One where I'd probably end up in a hotel. Threatening my job is never acceptable. (Thankfully, my wife would never do that.)