I think this will depend on what style of balusters you go with

Your staircase has a closed stringer.

Options for staircases with closed stringers

rather than worry about the spindles and rail base, I would remove all the carpeting.

if you need it, a runner would be attractive and just paint the risers and skirt board the same color you choose for the trim. the existing style is consistent with the door style, it all depends on how much style you are changing throughout the house.

If you remove the carpeting, you'll likely find rough plywood stairs. Nothing you can use with a runner without changing out all the treads and refacing all the risers. It will be $$$

Selecting a very flat, level loop commercial carpet for the stairs would look much better than the cut pile currently installed.

A carpet like this but running from stringer to stringer

I do agree that carpet needs to go but redoing stairs can be a big expense so at least change the runner to something more practical and no carpet on the side wall ever. Choosing the balusters is a job that we can't help with until we see your style

We’re definitely removing the carpet! No worries there. Thank you for using the term “closed stringer” that gives me a better idea of what I need to be searching for!