We have an antique pine island with our main sink. It's finished with Good Stuff for Wood. We've been in our home over 7 years. I do wipe up any splashes when I'm done - don't leave water sitting long periods or overnight. We haven't regretted using wood with a sink.

We have a prep sink in new pine on the coffee station/black hutch, linked below.

Here is a link that might be useful: French Gray Island kitchen

The countertops in our kitchen are oak butcherblock, and we have two sinks. It's been a year and a half and there have been no problems. Both sinks are topmounted stainless steel. The main sink has a wide ledge at the back where the faucet sits, so that faucet is not directly against the wood. The secondary sink faucet is sitting on the wood. The countertops came from Ikea and are finished in their Behandla oil, which is linseed based. I've been very happy with them. We wipe up water spills and don't leave wet things sitting on the counters too long. Also the faucet handles are oriented forward and we use motion-sensor soap pumps. I didn't plan those on purpose to prevent drips onto the counter, but it turns out they do, so that's handy.

I put under mounts in wood tops often. I have the edge and part of the bottom epoxied though most often the rest is an oil. I will not do it in red oak as that species is like a bundle of straws so can more easily pull in moisture. White oak OTOH has superior moisture resistance.
I use a company called Engrain most of the time, not inexpensive but worth it IMO.

This one was made of hard maple, with a Silgranite Super Single UM sink. I have also done them in Cherry and Black Walnut. I've never had an issue with any of them.

We usually finish all exposed end grain, with West System 2 part epoxy, and the rest in oil.

We have an undermount prep sink with a walnut countertop. It is my belief that the people who say not to do this, have no experience with wood countertops. We have always heard that water will damage wood. Yes it does, but it will also damage other surfaces. Our entire kitchen would have been wood counters, but DH said no. Another person that just did not think a wood counter would work in a kitchen.

PS - I have wood floors in a bathroom with no problems. Rebel!

We have an undermounted prep sink in beech butcher block from Ikea, and it has been fine for over 4 years. We had to reseal the rim where the sink meets the wood a couple times, but finally used Polyseamseal and that seems to be working much better. This sink gets heavy use, as it is where we have installed the filtered water faucet.

We have a positive reveal on our sink, so we do wipe it off when water pools there. Since the sealing with the "good stuff", over a year ago, we have not had any issues of separation, even through the seasonal changes where wood contracts and expands. Our breakfast bar gets plenty of use and abuse, though we do not cut directly on it. We finished it with Waterlox, and I can see some spots that need a little touch-up, as we had a few incidents with permanent marker. :/

If you are looking for reclaimed lumber try real antique wood in Irvington. Not the nicest location, I was just there, but they have nice reclaimed wood for floors furniture etc and they will make an island top. Good luck.