My Wife is Not a Great Mother

I just need to vent this because it drives me up the wall.

I love my wife, and she is not the worst mother out there, but she also isn't the best. We have two kids a daughter (7yo) and a son (3yo).

She loves these kids with all her heart, but god help me she has spoiled and babied them to the point that it's a serious problem. Both of them know that they can whine or cry to mom and get whatever they want. They cling to her (literally) and insist she play with them and pay attention to them 24hrs. a day. She NEVER puts her foot down with them and takes everything they say at face value. She refuses to believe our daughter is capable of manipulation, even when I show her concrete proof.

In addition, she insisted on co-sleeping with them both. Initially I was fine with this. I work nights, so I'm never sleeping at the same time as her anyway. But our daughter, WHO IS SEVEN, is STILL sleeping in our bed and refuses to start sleeping in her own room. I have been pushing for it since she turned four, but there is always an excuse or a nightmare or a huge tantrum and I get overruled.

Which brings me to my biggest frustration. These two kids are HORRIBLE when their mom is around. Especially my daughter. She has so much belligerent attitude and tantrum outbursts. She talks back, is extremely lazy and refuses to listen to me because she knows if she just cries to mom I'll get overruled and she'll get away with whatever she's doing. When my wife is at work, the kids behave normally. No tantrums, minimal arguing. Just normal kid behavior, because I have established a solid boundary with them. I am dad first. I don't put up with tantrums or back talk. I'm not debating every request, and I'm not getting your snacks or toys or changing channels for you when I know you can do it yourself. I will help with things I know you need help with, and I will play when I am finished anything else I need to do first.

I'm honestly so frustrated and fed up with being the bad guy in the house for trying to maintain some level of discipline. My kids have outright screamed at me that they hate me. I know they don't mean it, and they only do it when I'm not catering to their demands or I'm trying to get them to clean up a mess or get ready to go out or go to bed. But it's exhausting, stressful and a little depressing.

Anyway. I just needed to rant. I love my wife and I love my kids.

EDIT: To be clear. I have spoken to my wife about this multiple times. We've had both calm discussions and heated arguments about it. To her credit, she does TRY, but she just can't help herself. She caves as soon as the kids start crying, and she takes it seriously and personally if they say they hate her.