You do not have room for an island in the kitchen. You have room for a breakfast table next to the kitchen

I was going to suggest an outdoor rug because with larger rug sizes I personally prefer to hose them off outside and let them dry in the sun instead of using my washing machine. In my last kitchen I had some larger rugs like that, I’d take them outside to my sunny pool deck, spray them as needed with a degreaser, then hose them off with a powerful spray. They lasted many years that way. But I got rid of them when my color scheme changed with my current kitchen.

For your kitchen, I would do a rug like you have drawn out. That said, I can testify DD2 had a killim in her last house/kitchen. Even with a rug pad, it did not want to stay down. They are not heavy rugs, and tend to "walk" and move about. I would suggest something heavier than your etsy rug. We've had rugs in our kitchen for decades. When the children all at home (and cats/dog), I had a printed sisal (ordered from Ballard Design). Also had one under the breakfast room table. No issues. Easy spill clean up and vacuuming a breeze. Last house, I decided no rugs in the kitchen, but used Soumak rug in laundry room (and many other rooms). Flat weave rugs, are virtually indestructible, and they stay down. Liquids bead, so easily cleaned. This includes cat vomit. :D Check Kalaty website (then shop around), as they make odd sizes some manufacturers do not make. I have three nice rugs in our current kitchen. 3 dogs, 2 cats and 2 grandchildren, but DH is my mess maker (arthritis). I have been buying rug pads from Rug Pad USA for years. There is a great buying guide to help get the right pad for your floor and rug types. They also custom cut to size. (Have you ever tried cutting a nice rug pad with scissors?!) BTW, for pet or food stains, keep the small bottles of club soda on hand. Get up spill/whatever as much as you can with a spoon (lifting it off, not getting it deeper). Then use the spoon to put just enough club soda to cover the spill. The bubbling lifts the spill to the surface, then firmly blot with paper towels. If needed, repeat. I usually put aluminum foil between rug and rug pad, so the rug pad (and flooring) doesn't get wet. This method has been great to remove cat vomit stains from the rugs in our current home (not just the kitchen). Replace top and shake bottle. Store in refrigerator until needed again, but if it doesn't bubble the next time you use, toss and open a new bottle. (Hence, buy the small 6 pack of club soda).

I got a ruggable to try it out in my kitchen... got the extra thick mat. I cannot use my Dyson vacuum on it (just stops). However, the rug washes very well. Mine sits on top of the padding. Both the pad/rug creep a little from our walking path on it, but it is easy to pull back into place (we have a 3x5, which may move more readily than a runner). I really like the thicker padding while standing at the sink (and we have wood floors).