Are wooden kitchen counters meant to be shaved (with a planer) to continue looking tidy?
No no no on the planer.  Unless you can totally remove the top and send it through an industrial planer that can do it all at once, a planer is a terrible idea.  It will just be full of ridges, you will have bad edges, and probably take off too much.

Normally I would hit something like this with 40 grit on an orbital sander - not belt - and then slowly work my way to 300-400 grit.  Oil it a few times over the next two weeks, and you are good to go (have rehabbed many butcher block counters).