https://ruggable.com/products/heathered-solid-chocolate-brown-rug?variant=31668174651447

I would use a ruggable rug. They are machine washable and look fantastic. Because of your busy pattern on the floor, which appears to have a lot of browns, I would choose the above rug.

LLBean has some great runners that are low pile and don't move around too much.

I don’t think I would add a runner and cover up the beautiful tile. Is it really necessary?

It's needed since it is a main thoroughfare from our garage into the house. Lots of kids coming in and out with dirty feet.

I would use a synthetic jute. They're indestructible. You can take them outside and hose them off if need be. This site lets you customize the exact size you need and choose a border color. https://www.sisalrugs.com/all-weather.html