This isn't going to be super-helpful because I can't remember the name of the product, but I had an experience using a gel based stripper that did pretty much what you described that you wanted: it turned the old finish into goop that I could just rub off with any kind of scratchy pad.
The old finish was a varnish and stain rather than paint, so maybe it's the paint that won't soften?
Before I committed to actually sanding the whole thing down, I might try a different product if you can. ETA: I do think that I had to sand before I re-finished, even with what I used, but it was just a light sanding to make sure you got the stripper stuff off, not the kind of sanding you'd have to do to actually consistently remove a finish all over.
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