IMo the easy answer it to rip it out and should have been done from the beginning adding another layer of frlooring on top will create a whole new set of issues like your DW won’t fit under the counter . Rip it out get some vinyl tile so much easier on feet and knees and do this right. Even if you could find the same tile the dye lots from then til now will make the tile look diferent anyway. IMO do it right the first time. Do you have 48” walkways allaround that island just out of curiosity and maybe you could remove the tile from under the island to replace the ones you need to replace .

I've found designers and architects to be helpful when someone asks about a particular design or piece of theirs. Have you written to the architect?

If you JUST bought a new home on 20 acres, I would think through who your visitors will be and how often they will visit. (You don’t have to disclose that here.) Some of us prefer to have few visitors. Many of my neighbors have gates to block people from driving onto the property. In general, no one around here with a large-ish property expects people to drive onto the property without an invitation. I get deliveries multiple times a week from fedex and ups. At least with fedex, you can create an account online and provide special instructions. Not sure about UPS and USPS. Any appliance or freight delivery services will schedule with you and ask for any special instructions. If family and friends are visiting you, you can text instructions to them, and they’ll get used to the routine. If you are having a lot of guests you don’t know, then you can think through the experience you want them to have. Signage is one option. Hardscaping is another. There are other options too. If dog containment is an issue, be sure to think through all of that if the dogs are left outside unsupervised. Where I am, there is a lot of wildlife and my dogs could easily take off after a scent. Fencing may not be enough for some dogs. Some dogs are territorial and may not like delivery people. There could be liabilities. There could be snakes that could harm the dogs. Think through all this before you decide to let your dogs have access to unannounced guests. Living on a large property is a lot different than living in a suburb on a 5000-10,000 sf lot. It’s possible that after you settle in for a year or two, your visitors might be predictable and accustomed to your property. Adjust the property to meet YOUR needs, and not those of your guests, unless you plan to hold a lot of events with first-time guests. If you bought the property for privacy, enjoy it! Privacy is addictive. But if people are coming and going all the time, even your ”backyard” may not be as private as it would be in a suburb, because it is not necessarily protected by a fence and the house.

Sorry I was focusing on the vertical element not the floor, which your question was about. I agree though that you should rip that white tile out. It is terribly dated and will be an eyesore once you have new material for the chimney and the floor around the fireplace. And whatever you put on the floor for a hearth will be yet a third material. Too much going on. So I would just do the wood in the kitchen unless you want tile in the kitchen, in which case you can find a kitchen tile which you can extend to use as a hearth tile too.