Hi,
Believe it or not, the best method for cleaning/resurfacing the butcher block is scraping. Any sandpaper will clog up with the grease that's penetrated the wood. Chef's supplies will sell a tool for the purpose, or you can use a paint scraper or a cabinet scraper. Used properly any of these will yield a surface ready for a new coat of mineral oil.
To refinish a handrail, maybe start with some denatured alcohol and steel wool, using a tiny amt. of liquid and working a few inches at a time, wiping up drips immediately; you can actually clean it without much protection below. If the alcohol fails to remove the finish, move up to a refinisher like Formby's. As little as I like this stuff (too $$$), I think it will fill the bill for you. The new finish could be shellac, or penetrating oil. Either one will avoid the plasticky look of poly that is so unbecoming for a vintage home.
Casey

"...start with some denatured alcohol..."

Start with paint thinner. It will not damage the vast majority of surfaces and removes grease and grime very well.
You can decide from there how bad the finish really is and decide how to proceed.

Shellac is very poor for banisters.
Deft clear wood finish (brushing lacquer) is very good.
Complicated shapes like banisters require a lot of hand work on the curves.
If the wood is stained it can be hard to work a small area without refinishing the entire piece.

I don't like Deft for any surface that gets skin oils on it....the oils react with the laquer and make a gummy finish. For a surface that gets lots of hands on, shellac or better yet varnish is a better idea.
I also second starting out with mineral spirits....but I recomment a mixture of shellac thinner ( alcohol) and laquer thinner (acetone) to remove a clear finish quickly. It just melts the finish and you can wipe it off with paper towels.
Linda C

Lacquer thinner will dissolve dried lacquer. It will clean most normal dirt, but will not dissolve varnish or shellac. Alcohol may dissolve dried shellac and will also clean normal dirt. Paint thinner will not dissolve varnish---varnish must be stripped/sanded to remove.

Many finishes have a shelf life---shellac rarely is much good after 2 years of sitting premixed. Oil finishes like tung oil or boiled linseed oil will not dry properly after as short a time as a year on a shelf, and remain gummy and sticky.

My preferrence for railings is to finish with varnish. It is long wearing, easily applied, and easily repaired if necessary. I buy interior varnish for customer jobs at a paint store---NOT a home improvement store.

I find shellac to get gummy from hand oils. I have never had or heard of a problem with real lacquer being affected by skin oils.
It is nitrocellulose plastic (the same stuff ping pong balls are made of) dissolved in a carrier and pretty immune to reacting with much of anything without the correct solvents around.
The only time I have found sticky banisters was when shellac was used. Moisture alone will cause shellac to soften (and if it gets wet enough blush white). Oils left on shellac can start to react with it.