Match the wood as best as you can. No sense in chopping flooring up, IMHO.

Your floors look to be classic solid oak - which makes it easy to match. While many of the newer floors are very pretty, they tend to change out the styles and after about 18 months or so it can be hard to find a match. This is a timeless, if not trendy floor and can be found in homes over a hundred years old - many of them very elegant. They will look great refinished.

They appear to be natural, but the finish has ambered over time as old oil base finishes tend to do. You can refinish and stain them to match new site finished oak floors. Natural or a light stain is pretty and flexible style wise. Very dark stains show dust and scratches more. It often does not go well to try to make them gray. It often doesn't look great, especially if they are red oak and it is a lot of money to spend for something that is trendy and will go out of fashion in a decade.

You can have your flooring person put some sample stains on some scrap pieces to help you make a decision. If you really like a stain, ask to have a few different pieces of wood samples done because there is a little variation from piece to piece and you want to get a sense of the lighter and darker tones of the same stain on different pieces.

I totally agree with Fori. A professional can make it all match. Stick with the light color. Avoid the trendy gray.

It might be a good idea to have a floor pro make some stains for you to look at as already suggested. But also make the same stain over new oak planks that will be used just to make sure there will not be too much of a contrast.

What Fori says. I had site finished oak floors matched, and there's no way I could see where old and new met. Contractor feathered (?) the old floor at room transitions.

When I had my 100-year-old oak floors refinished a few years ago, the guy had to replace some of the boards, and you seriously cannot tell. In case you needed another voice to tell you matching should not be a problem.

I would install hardwood in the new areas and have it match. I'd actually consider having the old refinished so it matches with the new. I would go with the same color or something similar. I wouldn't pick anything trendy (gray or really dark). The less different flooring there is the less visually choppy a house looks, IMO.
You could ask a flooring guy about the wood and color but I think it looks like natural oak without stain and it has ambered over time.

I wouldn't do the vinyl.

I agree. It looks like oil based polyurethane that has yellowed. Check out water based poly for the refinish. Less smell and looks more natural over time.

Yes, a professional will be able to match your existing floors. I would also refinish the old as long as you are doing flooring.