AITAH for getting upset that my wife isn't being supportive about a punishment for our child?

For context... my wife and I are really bad at communication. It's something I'm trying to remedy, but that's a topic for another post.. also, this may come out as a word salad because it has escalated quite a bit from the original topic.

I (34m) am conflicted with my wifes (33f) response to how I chose to handle a situation with our son.

Last night our son (10m) went somewhere he knew he wasn't allowed, and got himself grounded. In the midst of explaining this to our son, he stormed off and before he closed his door he yelled "Fuck you, Dad!" I calmly walked upstairs, took his phone and explained that he would be grounded another day for an outburst like that.

His mom was downstairs when it all happened so I went down and explained to her his punishment and the extent thereof. She barely acknowledged me, which I knew would be a problem later but I wrote it off anyway.

Fast forward to this morning when I get a phone call from my wife stating our son will be at his friends house or the park, going directly against his punishment. I got upset, and explained to my wife that I don't feel like she has supported me in this situation and is basically reinforcing my sons sentiment "fuck you, dad." Furthermore I feel like it gives my son the idea that what dad says doesn't really apply as long as mom gives him what he wants. He isn't going to learn anything from this situation.

Not only that, but after I expressed my feelings to my wife, telling her I feel like a really bad parent and explaining that some positive reinforcement from her to me in that situation would have been really helpful. A simple "you're not a bad dad, it's just a bad moment" would have changed the whole situation in the moment... she proceeded to tell me she isn't here to coddle me and my feelings are mine unto myself. She doesn't have the capacity to give me any kind of words of affirmation when I need it and I should probably look for someone else to do that, because she's not my mom. The conversation basically ended with her saying she's going to be looking for an apartment, and we are going to be splitting up.

Am I over reacting to this whole situation? Should I have just let it go and not been so upset about what my son said and how my wife reacted?

Tl:dr; son said "fuck you dad", dad grounded him, mom let him go outside and play with friends anyway. Am I over reacting thinking I'm not being supported in this situation by my partner?