Not everyone can accommodate an kitchen island large enough for a small dance floor or sixty linear feet of perimeter cabinets. If space is a premium, it's obvious that all the area available has to be utilized completely. In that situation the use of multiple countertop materials may not be either practical or even desirable from a design standpoint.
Granite and other stones aren't suitable for food prep that involves direct chopping. They dull knives almost instantly in addition to being damaged in the process. Synthetic materials have some of the same issues. Wood is user friendly in so many ways. If the correct sealer is used it can be used for chopping, rolling dough and every other kitchen chore in the book.
One thing that makes wood counters unique is the way they age gracefully and gain character through use. Especially if sealed with an oil or wax as opposed to a varnish sealer, wood wears all the little dings and dents like a grand old lady wears her good jewelry.
This tiny space is charming and as functional as possible. No room for designated areas here so all the space has to used.
Small spaces have issues with clutter. Lots of different materials can become visual clutter and a kitchen is especially prone to this. Keeping the color scheme simple and limiting the textures will make for a more spacious appearance.
In a very small space everything is in line for up close viewing. There isn't a lot of anything so rather than dazzle with size use the most beautiful and luxurious material you can. Invest in quality as quantity isn't an option.
Wood is known to have antibacterial factors and is proven to be as safe and sanitary as other cutting surfaces like plastic. So using your countertop for actual food prep is a great way to make the best use of limited space.
It can be a blessing in disguise to be able to reach anything in the kitchen in a step or two. Just imagine all the time you won't be polishing that bowling alley sized island top that looks so great in photos. Use the best material you can and enjoy having a space that works hard for you while looking good. Happiness is definitely an appreciation of the smaller things in life.