I did it to ours more than 10 years ago. It was time consuming but never any regrets. Here is how I did it. Lightly sanded the varnish. Dusted thoroughly. Primed Painted 2 coats of oil-based semi gloss. I used oil base because it provided longer work time to correct paint runs.

Be sure to sell or give away that beautiful old railing and not throw it on a heap. I happen to like it. But indeed, it might look better in black. The railing you have posted instead gave me some shudders. Quite a contrary style and not one I'd welcome in my home. Way too modern for a ranch home, is what comes to my mind. If you are really going to get a new railing, I like the one Judy proposed, although I wouldn't want it in white at your place. Perhaps in black and the railing that matches the wood of the stairs? Or maybe something like this?

Were it mine, I'd consider having the one single railing on the right side, and on the left side I'd have either the existing two segments of railing or remove the upper section along the wall as no longer needed. Re the left newel post being too tall, I think the installer needs to verify that the railing height is adequate. The railing appears too low to me. I am not sure you understand the complexity of the installation nor the costs involved. I am of above average handyman skills, but I would never tackle a project such as this. Likely costs are in the thousands.

I always like the the handrail to be slightly darker than the floor. In general this looks better for stained rails. In your case- either paint it darker and consider matching the iron as well so they don't compete with each other. OR you can consider a metallic paint for the whole system. https://www.modernmasters.com/landing/homeowners/brands/mpc