The problems with getting the finish stripped are puzzling and I'm not sure what an unnatural stain is. There just weren't that many different finishes 100 years ago. There are different ways of applying colour to the wood - as a layer on the surface (a varnish or shellac applied over the top), a pigment stain that is wiped on over the top and sits as particles in the grain, or a penetrating stain usually carried in a solvent, that soaks into the wood. Chemical stripping will remove varnish, it may remove some of the surface pigment and its unlikely to have much effect on a spirit stain. That means that chemical stripping is not necessarily going to get the stain out of the wood and you will have to mechanically remove the stained timber (by sanding or scraping) or use the right solvent. The legs looked very dark to start with and getting them to a point where they can be stained all the same (lighter and consistent) colour will be a bit of an exercise.
Leaving the table top loose is an option, but I'd suggest some short dowels in the top frame into the loose table top to stop it from sliding around. I doubt that felt on the inside of the drawer will help much, you might have to more or less dismantle the drawer to get a very neat result and I'd recommend not doing that.
Personally I'd get the legs resolved before thinking too much about expanding the scope of the project.
I’m also interested in people’s thoughts on the following.
I think one idea with antique refurbishing is to stay true to the original. For example, if it was possible just to clean/remove grime from this table - that would have been enough. Alas, I am stripping back more because of the ugly, hard to remove and ‘unnatural’ stains.
However I have another idea too. As the table has no markings or obvious link to a date not a person who built, it I am thinking of a few other refinements too.
For example, I could apply felt to the underside of the top so that at any given time I can use the top as a board without the table. Further, I thought I’d felt the inside of the drawer too.
I feel those changes would add a level of refinement and indeed, functionality.
That said, philosophically - is there a right and wrong way to approach such project with antiques? Am I in some way devaluing or disrespecting the past in making such changes?