If you use an 'inset sink' rather than 'under mount' sink this would allow generous sealing under and around the sink.
I would discuss this idea with a timber mill company first as you may find the timber will need to be rested for sometime before its settled enough to use in this way ... Lovely idea though !!

7288, please keep us updated on what you find out from the companies, and photos of your new sink and counter. :)

Here's a discussion about wood counters that I think will be helpful:
https://www.houzz.com/discussions/nice-wood-counter-top-type-of-wood-please-pros-cons-of-wood-vs-stone-dsvw-vd~134228

I appreciate your answers - thank you very much.