I'd keep the carpet only in the middle but add wood on both sides. You can have wood on the landings if you want.

GD,

I already know it will be expensive if we do anything other than a straight replacement of carpet. But thank you for pointing that out in case there are any other readers here who might wrongly think this is a cheap update.

C Martin, thank you. I'll talk to the contractor about that. Here is a wooden top landing in a HOUZZ picture, and then they transition to carpet for the bedrooms.

Here is a picture of the middle landings with wood:

In this picture, it looks like the carpet is not a runner over wood treads, but is set in between the side wood pieces. This looks nice to me. Do you think this kind of idea would work on my staircase?

Inserting carpet is twice the work, with two different installers. You have open stringer stairs, so you will already be rebuilding more than half the stairs to do wooden treads correctly. All the existing treads and railings have to come out. You might as well get your money’s worth out of that effort and have it all be wood. .

Wood with wood or painted risers looks the best IMO. Carpet winds up looking worn and dirty and matted. I run up and down my wood stairs all day long and don't find them slippery.

We have wood stairs and they aren’t particularly slippery. I agree that carpet can start to look ratty on stairs. Our risers are painted.

I'm not sure where the impression comes from that wood treads are slippery. Carpet installers maybe?

It's not my experience at all.

Thanks for all this information. I’ll be meeting with the contractor soon.

Question: what organization or publication can I look at to get familiar with the building rules for staircases? For example I read the TCNA handbook regarding my shower, but I don't know where to go for a staircase.

I went onto nahb and icc but I'm just a homeowner so I don't know where to look.

looks to me that there are wood treads beneath all that carpet... Rip it off, and see how it works for you.
Wear shoes, slippers, or nubby socks on wood treads. I have not slipped on ours. But, I vuse at least one handrail, too.

personally, I don't love carpet on stairs.

Sorry to say, there are no nice wood treads under there. We saw the underneath when we replaced the carpet at move-in many years ago.

Does anyone have a link to building standards for stairs?

Where I live, stair dimensions are dictated by state codes. Here's an article addressing general rules: https://www.hunker.com/13402630/florida-building-codes-for-residential-stairs