Generally speaking, what is the most unobtrusive finish?

I'd agree that BLO or Tung Oil would be good and not add much color.
Mineral oil would probably add even less color, but since it doesn't
polymerize, it wouldn't protect the surface of the wood at all (except
maybe from drying out). If you need a more durable finish, though,
such as for chair arms, table tops, step stools, or anything that gets
used a lot, some kind of hardening varnish/poly would help a lot. You
mention a plasticy look with poly - have you been using water base
poly? I haven't used waterbase, but from what I've seen and read, it
adds more of a blueish tint or plasticy feel than oil base poly or
varnish. I've had very good luck with Rockler's poly gel finish - easy
to apply, protects well, and doesn't add any noticeable color. If it's
too glossy, you can knock it down as much as you want with 0000 steel
wool. Super-blonde shellac is another "unobtrusive" option - better if
you make it up yourself, but it goes over or under any other finish,
and again, you can control the gloss with steel wool or ultra-fine
sandpaper. Not quite as durable as poly/varnish.
Good luck,
Andy