I've gone through your posts and responded to each question accordingly...
...The top is butcher block, very old and distressed...
The top technically is not "butcher block" at all. That term is thrown around a great deal by dealers "selling" these, but with little knowledge of there proper treatment, what they are made of, or how to re-purpose them properly.
From what I can glean from just your photos, and my experience with very similar reclaimed materials is this is (most likely?) quarter sawn Douglas Fir wood stripping. It may have been already re-purposed once from an old bowling alley floor, as the field floors of the lanes (especially out west) are often a Pine species and/or Douglas Fir. This re-purposing for factory and industrial application was common between WWI and just after WWII. The factories that made the floors also did just make the long bench tops themselves so its hard to really tell without a full providence of the piece and its circa dates.
The give away on this top and its history is the end grain patterning (very tight and likely between 1900 and 1930 in orgin?) as well as the very common (but unusual outside this application) joinery system employed for this type of wood stripping assembly method. Its vernacular name is "alley T&G" or "ally joint" for the slang. It is a type of tongue and grove joinery system.
Below photo is a more contemporary version of it:
...I intend to use it as my kitchen island, and it needs a finish that I can wipe down daily. The wood is oil stained in places, has some gouges and pits and small holes. I have sanded on it, but I can't completely get rid of the splinters. Each time I get through a section, new splinters form...
The "splintering" is common for this species.
As you are learning, just sanding will only reveal more and more interstitial distress fractures deeper in the top.
... I don't want to lose the distressed look, so I'm looking for the best way to preserve that while giving a smooth finish...
There are several available to you. Some are contemporary "film finishes" that I DO NOT recommend nor advise on. The others (traditional oil-wax-rosin blends) are germane to not only the industry, but also your intended used as a food prep surface.
...I don't want to fill the holes and gouges with putty, so I was leaning toward an epoxy bar top finish, but a high-gloss finish is not my preferred flavor for this project...
I have used epoxies for such projects, but this is not something I typically recommend unless I'm doing the work myself to those without experience with these very specific finishing methods. Epoxy work is often (very often!!!) not done well and its hare to see that until it's too late.
I would go with natural finish from a company like Heritage Finishes
or The Real Milk Paint Company.
I've worked with both for over 20 years. These small companies make a living from traditional/natural finishes, love their craft and art and love helping folks. If you are set on an epoxy...(not recommend at all as it is typically used)...I do work in it for some applications (vivarium, abattoir floor/walls, etc.)...and then I go with EcoPoxy
however the application modalities are very specific in focus and application for both means, and material chosen. I will not give that information out publicly so you would have to email me.
...I have read about methods of sanding the finished epoxy with 1500 grit and then waxing to produce a more satin finish, and also covering the finished epoxy with a satin polyurethane. Does anyone have experience with either of these, or is there a different process I should consider?
Experience?...YES!...quite a bit, and neither "waxing" nor "Polyurethanes" are recommended at all in your project's goal parameters.
...If I decide to go with the epoxy, will the oil-stained sections cause the coating to fail?...
Very likely and a common issue with such projects. Again, you will have to email me for details because if you are going to use an epoxy, this top needs a poultice cleansing be applied to the stained areas and I do not share that info publicly.
DO NOT APPLY A POLYURETHANE!!!
...I don't want the BBQ Joint look either. When you said to fill the holes with epoxy, do you mean just spreading it on with a putty knife or squeegee until the surface is flat? My next question would be, how long should I wait before applying the polyurethane?
I also do not recommend any form of "surface flooding" for this type of food prep surface nor in general.
... Do I need to look for something that's "food safe"?
YES!!! not all "plastic film finish" are safe. They may cure, yet not react well to citrus, and other food oils.
...This afternoon I'll start with the wood hardener. I'll definitely keep you updated!
This too may not have been food safe. Some are, some are not...
...I think you can brush the epoxy on, rather than "flooding" it on, and it won't give the appearance of the plastic layer. And from what I've read, you can put a poly coat on top of that to get the sheen you want. I dunno. I'm going to try on a crusty old board too, before I commit.
No flooding is recommend nor the poly as stated above.
Look forward to an email if any of the above is to your interest.