@bamboo34 Sleep deprivation is absolutely horrendous. Looking after the baby all day is a hard job when you are sleep deprived. She needs a break from baby for a little rest and recreation.
What you listed were pretty standard parenting responsibilities. Along with changing nappies (which you forgot to mention).
She is the main caregiver, she will be basing routines, and household parenting ethos on baby’s needs that she has learned from her observations, experiences with baby and knowledge gained from spending all day and night with baby. So, respect that.
Have you helped your partner by doing nice things for her such as making her a cup of tea, giving her a hug, telling her that she is a fantastic mum, showing her appreciation, making sure that you both get equal downtime, listening to her, etc.
And FWIW separation anxiety is absolutely normal. Being clingy with mummy, absolutely normal.
My dd when she was a baby didn’t like me going to the toilet without her. When her dad was home from work. He’d hold her while she was crying in the bathroom reassuring her that see mummy’s on the toilet. She’s still here. It’s ok. She’s much older now. They have a great bond. Responding to baby’s needs develops trust and really pays off.
She went through a stage of preferring mummy then a stage of preferring daddy. Then back and forth on that. Perfectly normal and neither of us took it to heart.
If your partner is stressed and exhausted, just show her some love and understanding. Kind gestures and good listening go a long way.