I've got half a newel post at the bottom of my stairs (minus the dado) where someone mucked around with the stairs adding a huge platform at the bottom with banister, spindles etc ....I've put off sorting it as I need to find matching parquet but if you've got the budget I would take the posts and dado out TBH. As forzaitalia said it's personal preference but I would paint the whole lot as shown below in a colour of your choice obviously!
So love your spacious hallway! I think if you paint it, it will be a great improvement. I would paint the skirting and dado rail too. It will brighten everything up a treat. Don’t just think white, sometimes a touch of dark looks stunning, and perhaps leaving the post newel and top rail natural can look very cosy. Here’s a few pics to inspire you