I am out of touch with nanny salaries (it has been a while) and I think expecting a nanny to be 'self employed' is a bit hopeful, but the job itself seems pretty standard.
My nannies ( who tended to stay long term, because they were well paid and appreciated) did all the bits of looking after my DCs that I wasn't around for or didn't want to do. So school runs, sorting out stuff for school, play dates after school and the dreaded homework featured largely.
When my DCs were small, I generally didn't get home until 8/9/10pm and wanted to spend what time I had with them doing things we enjoyed, not the stuff that could be outsourced.
Oh and I had a cleaner, a gardener and someone who did the ironing. It made sense, because it meant I could focus on my work and on enjoying the time I was able to spend with my DCs.