I am 99% certain that is rift sawn white oak. I can tell it is rift sawn as opposed to quarter sawn because I don’t see any of the “rays” that are typically present in a quarter sawn oak. I believe it’s white oak because the yellowness indicates such where as red oak is more white but with red/pinkish grain hence the name. There are testing kits that use wood shavings to test exactly what species of wood grain it is.

We’ve had our fair share of white oak floor repairs and one thing I will say is that because of the age of the treads, getting an exact match, colorwise, won’t be possible because of the difference between old growth and new growth and the aging your treads were subject to. For the most part the difference in color should be subtle but could be noticeable.

If you do go ahead and refinish these steps, you could test the color matching by getting a sample quarter sawn plank, put a coat of poly on it, then sand a spot on the steps (ideally where the finished has mostly worn off so it’s easy to sand), coat with poly and compare the colors. If the color is close enough and doesn’t bother your then go ahead and replace the tread. If it stands out and bothers you(which I totally understand), I’d maybe consider doing a glue and epoxy repair if the split in the tread is clean enough for you to get a tight and sturdy fit with glue.

I do see what you mean with the top tread. Looking at the wear on the lower stairs it looks like there was a was a runner on the steps and it must’ve been glued down at the top. Lucky for you, a good sanding with 40 or 50 grit will clean those steps up real nice and the color of all of them should be consistent.

Floor Refinisher here that has experience refinishing stairs.

Id say assess all of the other treads for signs of instability like splits or creeks. If nothing stands out then you I would say those steps are in good shape.

From what I can tell from the pictures you showed, it seems like you have either quarter or rift sawn oak treads. If you could upload a top view picture of the grain I could confirm whether it is. If it is, that combined with the potential age of the steps makes those treads priceless. Of all the floors I’ve seen out there, I think there really is no prettier grain than quarter sawn oak. I would absolutely preserve them if you can.

As some one who has refinished many staircases, I will tell you that those stairs are an excellent candidate for refinishing, and if you’re looking to DIY the job, that stair case is about as easy as it gets for stairs (floating railing and no wall strings makes it very easy to sand the treads with the edger and makes it much easier to throughly scrape the bullnoses.)

I would consider stripping the varnish on the risers and painting them white and perhaps add a wall stringer for a more finished look. That combined with a natural finish on the treads and a repainted railing and you have a gorgeous and timeless staircase