Suggestions for Cleaning 60 yr old varnished/shellacked Kit cabinets

Try scrubbing with TSP. This will take off a lot of dirt and grease.

I just answered a similar post.

https://www.hometalk.com/diy/paint/furniture/q-how-do-i-get-a-better-look-with-these-cabinets-unsure-of-coating-37348624

I really like using a deglosser and steel wool, but you have to make sure you use a damp towel to remove all the residue, and steel bits. Use a medium-fine steel wool to start

This is my 4th attempt to answer you...It will not accept my comments...

Whitewashing

A product called TSP is popular in cleaning kitchen cabinets before painting them. For removing the finish, another product called Citristrip is a popular choice.

These look so familiar! You need a good degreaser with vinegar, a scrub brush, and towels or something as a drop cloth. If you can remove the doors and hardware first, it will be easier to clean than mounted cabinets, but do what you have to. As you wet the cabinets, let the degreaser (Dawn, Mr. Clean, Ajax, etc.) soak into the highly dirty areas just a couple of minutes, and expect some mess as you clean off sixty years of dust, dirt, oil, grease, food spills, and varnish. TSP is also good with varnished wood. You may have to scrub the cabinets more than once, but you want them clean before you sand, stain/paint, and seal the cabinets. Best wishes ☺️