First a question: Why do you want to replace this custom railing? Based on your photos, it appears to be in good condition. Perhaps the railing needs to be refinished and the balusters need to be painted, but is there another problem? If posts are loose, for example, this can be fixed.
Second - rail parts: With the custom curve and joints, this will be a time consuming project for the contractor - if they do a good job. Each of the parts needs to be carefully fit together, and the joints need to be sanded smooth See this supplier as an example:
https://www.jmpwood.com/collections/handrail-fittings (Hence the reason for my initial question.)
Third - treads: Are the treads hardwood? If so, they should probably be sanded smooth, stained to match the adjacent floor and landing, and finished with polyurethane. If the treads are not hardwood, they should be painted with anti-scuff 'floor' paint - and
not white. You should try to match the color of the adjacent flooring and landing. Here is an example of the type of paint to use: PPG1240-5 Sorcerer Satin Interior/Exterior Floor and Porch Paint. Benjamin Moore and other suppliers make similar paint. You should spec the type and brand of paint to use, to ensure quality.
Fourth - risers: The risers should be painted white. Here is a tip: use special 'whiteboard' / 'dry erase' paint. This makes it much easier to remove scuff marks. For example:
ReMARKable Whiteboard Paint and
https://www.lowes.com/pd/Krylon-White-Gloss-Dry-Erase-Latex-Enamel-Interior-Paint-Actual-Net-Contents-29-fl-oz/1000211721
Fifth - safety: Your stairs will not be safe while the project is underway, particularly because there is no interior handrail along the wall. If you cannot live downstairs, it would be very wise to take a short vacation or stay in a local hotel until the new railing is complete. Even if the contractor is insured and the insurance company will pay a claim, death or serious long-term injury could result from a fall.
Sixth - bid price: The bid is fair for the amount of work involved - although you should spec the details noted above. Before you accept a bid for this detailed woodwork, however, you should ask to see previous wood railing or custom cabinet work the contractor has done.
I hope all of this info is helpful.
