Short of pulling everything out and starting over, I think your flooring guys have the right idea. The two different wood species, however, will make the same stain look different, so the darker the stain, the less likely you'll see the differences. I would get your flooring contractor to put the same stain samples on a piece of white oak and a piece of red oak to see if you're ok with the variation.

If you like funky character for you flooring (there are some interiors that highlight this sort of floor) then feel free to repair the damage (termite damage is not something you want to leave in place) and then have a look at some different finish options.

Your flooring guys are correct - if you want a floor that HIDES what it is. But you are actually looking to "live with" the character as-is. That's a different approach.

The red and white oak will ALWAYS look different. Always. The darker you go, the more it blends. It is possible to get a mid-tone...or a dark mid-tone to help disguise the BIG difference in colour. You will still have a funky floor but it will be less distracting to visitors.

I'm wondering if a Rubio Monocoat in a neat stain/colour would do what you want your floor to do = look good while retaining the history/character of the wood.

You will have different colours with the red and white oak. That won't change unless you go SUPER dark. You can lessen the impact but working with a mid-tone brown or a funky mix. It can be done....but you need someone who thinks outside the box and who is a WIZARD with stains.

Send some photos of what you have and maybe someone will chime in with more specific advice.