Choosing American White Oak vs European White Oak is a matter of personal preference. American White Oak has tighter knots but fewer tannins. Also, the available grades can vary, so that will affect what you want. Suggest you go to a local showroom and compare them. If your project is in the United States, consider that purchasing products that are made in North America rather than Europe will reduce the carbon footprint of your project. And, if you purchase American, you're supporting the U.S. labor force.

Just a little Heads up. Brazilian Oak is called Tauri. It is an 'exotic' hardwood (it comes from somewhere other than your continent). Like MOST exotic hardwoods it is photosensitive. That is to say it GETS DARKER as it ages. This has NOTHING to do with the finish. It has everything to do with the WOOD. The pretty, soft, velvety browns and smooth beige you are seeing WILL NOT last. Nope. They get darker and STRONGER in colour. Yah. Even though they are ranked as 'low' on the photosensitivity radar...they will still get a snick darker as soon as they are exposed to 'light'. The white oak is not something that has this issue. In fact it can fade in a medium amount of time in direct sunlight (with old windows with very little UV block on them). And the STAIRS are going to be a NIGHTMARE with Tauri. The amount of effort put into 'matching' the colouring with another species can double your flooring budget. Nope. Not kidding. And just so you know...there are PLENTY OF COMPAINTS about Shaw wood flooring products. I would suggest you go ahead and search this website to see what people are complaining about. Some of the treads are old...and some are very new (like 1-2 weeks old?) which tells me Shaw is NOT pulling up it's socks on this issue.

Does your floor manufacturer make a matching stair cladding? If you do that, you don’t need to have oak treads. Another option is to do a beautiful contrasting color of stain on the stairs. You could pull the colors together with handrail or baluster or post colors. Adding a runner on the stairs, or painting the risers would give you more options to tie in the floor colors. If you really want to match wood, dark stain is worth a try, but lighter colors are really tough. I matched a red oak transition strip to a dark chocolate brown stone tile once. I had to keep mixing and adding stain. It worked only because it was a dark brown color. It’s hard to neutralize red and yellow tones.

@millworkman Thank you for your feedback. I know they will not look the same as they are different woods, but I am hoping to find something that will get me that lighter color with the least undertones if that makes sense.