Hi SSO.
Sorry to hear the finish is giving you problems. Before you reach for that can of stripper, you may want to try something else that might possibly work: Re-amalgamation & overcoating in clear shellac. While the top coats were originally varnish, the under coats were almost certainly shellac.
Please excuse me if I cover any pathetically obvious stuff, but here's what I'd do:
Use 0000 steel wool to introduce the barest amount of abrasion to the surface. (very, very light pressure -just enough to put some "tooth" in the very shiny spots) Then use paper towels and mineral spirits (paint thinner) to get off any residual oils.
After that dries, paint surface with denatured alcohol. (Not too much alcohol.) Work your brush slightly starting to move the original finish. It will probably start to "blush-up" stain and particles of filler. It will look pretty nasty. (This is where prayer and some more shellac comes in.) After this application dries (about an hour or so in a heated environment or longer in the cold garage), over-coat in clear shellac. Use the Zinsser in the yellow can (not spray). You'll want to use it slightly thinned with denatured alcohol.
Work in thin coats and let dry before next application. If you're lucky, it will start looking better with each application. You can retouch thin spots between coats with stain or even marking pens.
I had an Edison Standard model D that I recently rescued this way. After the first alcohol treatment it looked so bad that I really gave up hope that it would work. I kept at it with the subsequent coats and it began to look better and better. It ended up taking about 20 thin coats (sanding with 320 between every 4th coat), but it looks great. It also has the added benefit of the original color. So, in your case, if worst comes to worst, it can still be stripped and refinished.
I know this is not what you set out to do, and I wouldn't blame you if you decided to do something else. However, you are now at a good place for experimenting since you have little to lose and much to gain. Whatever you end up doing; good luck and keep us posted. (This way; if you learn something - we all learn something.)
Best Regards,
Martin