Yes, there are certain things that can help support their transition to independence.
I would say that at about 4, it is helpful for children to begin to spend short periods of time away from parents so that they can learn to negotiate with other children, start to develop supportive relationship with other children, spend extended periods of time concentrating on self chosen activity etc. without having a parent over their shoulder .
Nurseries also provide the opportunity for children to spend time in small groups with an adult (ratios are 1:8 in a private nursery or 1:13 in an LEA nursery (I think)) before they go into school where the ratio can be as much as 1:30 (unless there is a TA).
I would also say that 2 sessions a week isn't really enough for her to get really settled in. It's a long time between sessions if you see what I mean.
Writing names isn't important at this age. They should be doing lots of open-ended play and exploration e.g. den building, digging, painting, clay/sculpture, investigating cause and effect, singing songs, playing with sounds and rhymes, listening to stories, role play, counting, measuring, water & sand etc. etc. Far too much to list.
However, if a nursery is not good, then I do think she's better off out of there. I spent some time working in a nursery which sounds like the one you're describing. The children were cared for, but not stimulated or supported to develop, which led to children being bored and negative behaviour being seen.