First, i wouldn't be surprised if you found the original wood floor waiting for you in the bedrooms you are gutting. however, in this house (see photos), we removed the old kitchen (which was just tile over subfloor) and had to lay new oak strip flooring to match. they refinished both old & new floor, and you cannot tell where the old kitchen was!!!
(in photo, old kitchen was at left bottom of photo.)

Thanks for your advice. There isn't red oak in that room since the rooms were an addition to the house. That picture looks great!

Yes, keep the same width in red oak. You may have to have the older floor sanded down and refinished at the same time. Your local hardwood floor pro can advise you.

Red oak is pretty common. The width will be the challenge. But any wood flooring shop (locally owned with a reputation...not a 'big box' outlet) worth its salt will be able to source it for you.

I also vote for continuing the oak. Right now strip oak flooring has a standard dimension of 2 1/4". You can use a transition piece the width of the door jamb...flush 1x6 cut to size or you can take small pieces of strip oak laid flush in a herringbone pattern between the door jambs...labor intense but a great looking change up to the standard. Just remain consistent with whatever approach you take.