Remove top padding from List

Firstly, I would say that GroupedListStyle is working as intended.

On iOS, the grouped list style displays a larger header and footer
than the plain style, which visually distances the members of
different sections.

You say you have tried this, but it does work for me (Xcode 12.5.1):

    List { ... }    
    .onAppear(perform: {
        UITableView.appearance().contentInset.top = -35
    })

You could also hide the list header, by using a ZStack with the List at the bottom of the stack and the Picker over the top. The Picker does have transparency, so you would also have to add an opaque view to act as background for the Picker.

var body: some View {
    NavigationView {
        ZStack(alignment: .top) {
            List { ... }
            .listStyle(.grouped)
            .padding(.top, 30)
            
            Color.white
                .frame(height: 65)
            
            Picker { ... }
            .pickerStyle(.segmented)
            .padding()
        }
        .navigationBarTitle("Add Task", displayMode: .inline)
    }
}

As far as I can see this just appears the same as PlainListStyle would do, but I assume you have a particular reason for wanting to use GroupedListStyle.