As usual, this is a matter of trade-offs. What sort of maintenance is best for you? Do you mind if stains and blemishes are harder to repair?
If you want closest to "natural" wood, then a poly finish won't darken the wood as much as any oil finish. This is just the nature of oil finishes.
But poly creates an actual finish; a protective covering over the surface. This has a downside in that you probably won't be able to match repairs. Generally, is is suggested to remove all the poly and reapply if you want a nice even look.
With oil finishes you not only can reapply to freshen up blemishes, you have to reapply as part of regular maintenance. Oil will almost always feather nicely into existing oil. This is just not the case with poly.
I suppose you could take a hybrid approach, and use poly on the less wearing surfaces that you want a good water seal on (like backsplashes) and oil on the hard-working surfaces that you expect to stain and scuff over time. They will look different, however.
Finally, as pointed out in the comments you have another less-obvious path you can take: just remove the existing finish you don't like, sand/scrape bare and then just leave it alone. It'll age beautifully, and give you the aesthetic you want of looking 100% "natural" and will obviously be "food-safe". As it wears and stains you scrape it flat to get a new surface. This is how industrial kitchens treat their wood surfaces.
Most any interior finish is "food safe" once cured. Just read the label -- it'll tell you if it isn't. There is discussion of finishes suitable for use around food in previous Q&A. I encourage you to search some of those for further detail.