IMO runners in kitchens are just a tripping hazard maybe a nice anti fatigue mat would be okay .It also is a good idea to find something washable and for sure not hand knotted wool and honestly no mat is the best with water and grease all just adding to the things that need washing.

I'm not young and have never tripped on a runner in all the kitchens I have owned.

Hand knotted wool runners work just fine in kitchens. They spot clean easily for most things and can always be sent out for cleaning if you spilled something awful on them. People have been using hand knotted wool rugs for centuries and good quality ones are extremely durable.

I have my sink in my island and like yours it is across from my cooktop. From toekick to toekick there is 59” of floors space. I opted to use a 4x6 rug for the space instead of a runner. The one in the first photo was wool but not a hand knotted one. I had it for more than 6 years and would spot clean it as needed. I also used a carpet cleaning machine on it a few times. The rug in the second picture is a made of polypropylene. So far I have just spot cleaned it.

No one has tripped my kitchen rugs in the past 12 years but I’m sure if I let the Medicare House Calls person come to my house for an assessment they would get on my case about that rug and the other ones I have in my hallways.

Do you have a picture of your space? I'd do one up to the full length of the island, and measure 3"-4" in from where the cabinets end, so you have good width but can still see floor. It probably would feel better on your feet to have it scooted more toward the sink, so your feet aren't halfway on rug. But probably looks better in pictures to have it centered. :) I love a wool rug in he kitchen, and you can find many off sizes of vintage rugs on Etsy. Here's mine that I found on Lulu & Georgia. I had a vintage wool from an estate sale in the kitchen before we remodeled, but it didn't fit afterward.

I think this one looks too narrow.

Too wide but I like the length; you'd be able to see it from the other side of the island. But it also looks thin--see how the edges are curled up a little bit--there's Patricia's tripping hazard.

This looks like a good size and material.

I have a Ruggable runner , you can throw in the washer, lies completely flat, no tripping hazard at all.